APPLICATIONS OF THE DEFINITE INTEGRAL

Size: px
Start display at page:

Download "APPLICATIONS OF THE DEFINITE INTEGRAL"

Transcription

1 APPLICATIONS OF THE DEFINITE INTEGRAL. Volume: Slicing, disks nd wshers.. Volumes by Slicing. Suppose solid object hs boundries extending from x =, to x = b, nd tht its cross-section in plne pssing through (x,, ) nd prllel to the yz-plne hs re A(x). To first order pproximtion, the volume of the slice of the object on the right to the plne of thickness x is then A(x) x so tht the volume of the solid is the limit of the elementl sum A(x) x. This being Riemnn sum, the volume is given by the formul V = A(x) dx. Exmple.. () The Pyrmid Aren in Memphis hs squre bse of side pproximtely 6 feet nd height of pproximtely 3 feet. Find the volume of the pyrmid with these mesurements. () Find the volume of sphere of rdius R... The Method of Disks. Suppose tht f(x) nd f is continuous on the intervl [, b]. Tke the region bounded by the curve y = f(x) nd the x-xis, for x b, nd revolve it bout the x-xis, generting solid. We cn find the volume of this solid by slicing it perpendiculr to the x-xis nd recognizing tht ech cross section is circulr disk of rdius r = f(x), we then hve tht the volume of the solid is V = π[f(x)] dx.

2 APPLICATIONS OF THE DEFINITE INTEGRAL Since the cross sections of such solid of revolution re ll disks, we refer to this method of finding volume s the method of disks. Exmple.. () Revolve the region under the curve y = x on the intervl [, 4] bout the x-xis nd find the volume of the resulting solid of revolution. () Find the volume of sphere of rdius R. (3) Find the volume of the solid generted by revolving the region bounded by y = x +, y =, x =, x = bout the x-xis. In similr wy, suppose tht g(y) nd g is continuous on the intervl [c, d]. Then, revolving the region bounded by the curve x = g(y) nd the y-xis, for c y d, bout the y-xis genertes solid. Notice tht the cross sections of the resulting solid of revolution re circulr disks of rdius r = g(y). All tht hs chnged here is tht we hve interchnged the roles of the vribles x nd y. The volume of the solid is then given by V = ˆ d c π[g(y)] dy. Exmple.3. () Find the volume of the solid resulting from revolving the region bounded by the curves y = 4 x nd y = from x = to x = 3 bout the y-xis. () Find the volume of the solid obtined by revolving the region bounded by the curves y = x 3 nd y-xis nd y = 8 bout the y-xis. (3) Find the volume of the solid generted by revolving the region bounded by y = 4 x, the positive x-xis, nd the positive y-xis, bout the y-xis..3. The Method of Wshers. One compliction tht occurs in computing volumes is tht the solid my hve cvity or hole in it. Another occurs when region is revolved bout line other thn the x-xis or the y-xis. For exmple, consider the solid obtined by revolving the region bounded by the grphs of y = 4 x, x = nd y = bout the x-xis, nd the line y =, respectively:

3 APPLICATIONS OF THE DEFINITE INTEGRAL 3 Exmple.4. () Let R be the region bounded by the grphs of y = 4 x, x = nd y =. Compute the volume of the solid formed by revolving R bout () the y-xis, (b) the x-xis, nd (c) the line y =. () Let R be the region bounded by y = 4 x nd y =. Find the volume of the solids obtined by revolving R bout ech of the following: () the y-xis, (b) the line y = 3, (c) the line y = 7, nd (d) the line x = 3. (3) Find the volume of the solid obtined by revolving the region R bounded by the curves () y = x nd y = x bout the x-xis, (b) y = x nd y = 8x bout the x-xis, (c) y = 6 x, y = bout the y =, (d) x = 4 y nd y-xis bout the x =, The technique used to solve the problems bove is slight generliztion of the method of disks nd is referred to s the method of wshers, since the cross sections of the solids look like wshers.. VOLUMES BY CYLINDRICAL SHELLS Let R denote the region bounded by the grph of y = f(x) nd the x-xis on the intervl [, b], where < < b nd f(x) on [, b]. If we revolve this region bout the y-xis, we get the solid shown below: If insted of tking cross section perpendiculr to the y-xis, we tke cross section perpendiculr to the x-xis, nd revolve it bout the y-xis, we get cylinder. Recll tht the re of cylinder is given by: A(x) = πrh, where r is the rdius of the cylinder nd h is the height of the cylinder. We cn see tht the rdius is the x coordinte of the point on the curve, nd the height is the y coordinte of the curve. Hence A(x) = πxy = πxf(x), Therefore the volume is given by πxf(x) dx.

4 4 APPLICATIONS OF THE DEFINITE INTEGRAL Remrk.. Note tht for given solid, the vrible of integrtion in the method of shells is exctly opposite tht of the method of wshers. So, your choice of integrtion vrible will determine which method you use. Exmple.. () Revolve the region bounded by the grphs of y = x nd y = x in the first qudrnt bout the y-xis. () Find the volume of the solid formed by revolving the region bounded by the grph of y = 4 x nd the x-xis bout the line x = 3. (3) Let R be the region bounded by the grphs of y = x, y = x nd y =. Compute the volume of the solid formed by revolving R bout the lines () y =, (b) y =, (c) x = 3. (4) Let R be the region bounded by the grphs of y = x(x ) nd the x-xis. Compute the volume of the solid formed by revolving R bout the y-xis. (5) Let R be the region bounded by the grphs of y = 4, x =, x = 4, y =. Compute x the volume of the solid formed by revolving R bout the y-xis. (6) Let R be the region bounded by the grphs of x = y, y =, x =. Compute the volume of the solid formed by revolving R bout the x-xis. (7) Let R be the region bounded by the grphs of y = x nd y = x in the first qudrnt. Compute the volume of the solid formed by revolving R bout the y-xis. (8) Set up n integrl for the volume of the solid tht results when the region bounded by the curve y = 3 + x x, the x-xis, nd the y-xis, is revolved bout () the x-xis, (b) the y-xis, (c) the line y =... Summry. We close this section with summry of strtegies for computing volumes of solids of revolution. Sketch the region to be revolved. Determine the vrible of integrtion (x if the region hs well-defined top nd bottom, y if the region hs well-defined left nd right boundries). Bsed on the xis of revolution nd the vrible of integrtion, determine the method (disks or wshers for x-integrtion bout horizontl xis or y-integrtion bout verticl xis, shells for x-integrtion bout verticl xis or y-integrtion bout horizontl xis). Lbel your picture with the inner nd outer rdii for disks or wshers; lbel the rdius nd height for cylindricl shells. Set up the integrl(s) nd evlute. 3. Arc length nd surfce re 3.. Arc Length. Let f(x) be continuous on [, b] nd differentible on (, b). Our im is to find the length of the curve y = f(x), x b. We begin by prtitioning the intervl [, b] into n equl pieces: = x < x < < x n = b, where x i x i = x = b for n ech i =,,, n. Between ech pir of djcent points on the curve, (x i, f(x i )) nd

5 APPLICATIONS OF THE DEFINITE INTEGRAL 5 (x i, f(x i )), we pproximte the rc length s i by the stright-line distnce between the two points. From the usul distnce formul, we hve s i d((x i, f(x i ) ), (x i, f(x i )) = (x i x i ) + [f(x i ) f(x i )]. Since f is continuous on ll of [, b] nd differentible on (, b), f is lso continuous on the subintervl [x i, x i ] nd is differentible on (x i, x i ). By the Men Vlue Theorem, we then hve f(x i ) f(x i ) = f (c i )(x i x i ), for some number c i (x i, x i ). This gives us the pproximtion s i (x i x i ) + [f(x i ) f(x i )] = (x i x i ) + [f (c i )(x i x i )] = + [f (c i )] (x i x i ) = + [f(c i )] x. Adding together the lengths of these n line segments, we get n pproximtion of the totl rc length, n s + [f (c i )] x. i= Notice tht s n gets lrger, this pproximtion should pproch the exct rc length, tht is, n s = lim + [f (c i )] x. n So, the rc length is given exctly by the definite integrl: i= whenever the limit exists. s = + [f (x)] dx, Exmple 3.. () A cble is to be hung between two poles of equl height tht re feet prt. It cn be shown tht such hnging cble ssumes the shpe of ctenry, the generl form of which is y = cosh x = (ex/ + e x/ ). In this cse, suppose tht the cble tkes the shpe of y = 5(e x/ + e x/ ), for x. How long is the cble? () Find the rc length of the curve y = 3 (x + ) 3 between x 3. (3) Find the length of the curve x = 3 y 4 3 3y 3 between y Surfce Are. One cn esily show tht the curved surfce re of the right circulr cone of bse rdius r nd slnt height l is A = πrl,

6 6 APPLICATIONS OF THE DEFINITE INTEGRAL nd so the curved surfce re of the frustum of the cone shown below is A = π(r + r )L. Now, suppose tht f is nonnegtive nd continuous on [, b] nd differentible on (, b). If we revolve the grph of y = f(x) bout the x-xis on the intervl [, b], we get the surfce of revolution seen below: We prtition [, b] into n mny pieces of equl size s we hve done so mny times. On ech subintervl, we cn pproximte the curve by the stright line segment joining the points (x i, f(x i )) nd (x i, f(x i )). Notice tht revolving this line segment round the x-xis genertes the frustum of cone. The surfce re of this frustum will give us n pproximtion to the ctul surfce re on the intervl [x i, x i ]. First, observe tht the slnt height of this frustum is L i = d((x i, f(x i )), (x i, f(x i ))) = (x i x i ) + [f(x i ) f(x i )],

7 APPLICATIONS OF THE DEFINITE INTEGRAL 7 from the usul distnce formul. Becuse of our ssumptions on f, we cn pply the Men Vlue Theorem, to obtin f(x i ) f(x i ) = f (c i )(x i x i ), for some number c i (x i, x i ). This gives us L i = (x i x i ) + [f(x i ) f(x i )] = + [f (c i )] (x i x i ). The surfce re S i of tht portion of the surfce on the intervl [x i, x i ] is pproximtely the surfce re of the frustum of the cone, S i π[f(x i ) + f(x i )] + [f (c i )] x πf(c i ) + [f (c i )] x. since if x is smll, f(x i ) + f(x i ) f(c i ). Repeting this rgument for ech subintervl [x i, x i ], i =,,, n, gives us n pproximtion to the totl surfce re S, n S πf(c i ) + [f (c i )] x. i= As n gets lrger, this pproximtion pproches the ctul surfce re, n S = lim πf(c i ) + [f (c i )] x. n Recognizing this s the limit of Riemnn sum gives us the integrl whenever the integrl exists. S = i= πf(x) + [f (x)] dx. Exmple 3.. () Find the re of the surfce obtined by revolving the curve y = 5 x, x 3 bout the x-xis. () Find the re of the surfce obtined by revolving the curve y = x, x bout the y-xis. (3) Find the re of the surfce obtined by revolving the curve x = y 3, y bout the y-xis. (4) Find the re of the surfce obtined by revolving the curve y = 6x, x bout the x-xis. (5) Find the re of the surfce obtined by revolving the curve y = x, x bout the y-xis.

8 8 APPLICATIONS OF THE DEFINITE INTEGRAL Answers.. () () ˆ R R Answers.. () () (3) ˆ R ˆ R ˆ 3 π( R x ) dx = 4πR3 3. ˆ 4 Answers ( 6 5 ) dx = π( x) dx = 8π. π( R x ) dx = 4πR3 3. π(x + ) dx = 8π Answers.3. () () (3) ˆ 8 ˆ 4 Answers.4. (b) (c) () () (b) (c) (d) (3) () (b) (c) (d) ˆ 4 π(y 3 ) dy = 96π π( 4 y) dy = 8π. ˆ 4 ˆ 4 ˆ π π () () ( π( 4 y) dy = 9π. ˆ ( 4 x ( ( 4 x π( 4y) dy = π. ) ) dx = 8π ) ) dx = 56π π( 4 y) dy = 8π. (( ( ) ) π 4 x ( 3)) 3 dx = 47π ( ( ) ) π 7 7 (4 x ) dx = 576π (( π 3 ( ( 4 y)) 3 ) ) 4 y dy = 64π. ( ) ) π x (x dx = π ) 8x ) ) π(( (x dx = 48π (( ( ) ) π 6 x ) dx = 83π 4 ( ) ) π(( y ( )) dy = 3π 3 + 4π.

9 Answers.. () () (3) () (4) (5) (6) (7) (b) (c) ˆ ˆ 4 ˆ ˆ ˆ APPLICATIONS OF THE DEFINITE INTEGRAL 9 πx(x x ) dx = π 6. π(3 x)(4 x ) dx = π 6 = 64π. ˆ ˆ ˆ π( y)(( y) y) dy = π 3. π(y ( ))(( y) y) dy = 8π 3. π(3 x)x dx + πx(x(x ) ) dx = π ( ) 4 πx dx = 4π. x πy(y ) dy = 8π. πx(( x ) x ) dx = π. (8) () Disk Method: (b) Shell Method: ˆ 3 (c) Wsher Method: ˆ ˆ 3 π(3 x)( x) dx = 4π. π(3 + x x ) dx = 53π πx(3 + x x ) dx = 45π. (( π 3 + x x ( )) ˆ 3 Answers 3.. () ˆ 3 () ( + x ) dx =. ˆ 8 ( 3 y (3) (e x/ 4 ) + dy = 9. 3 y ˆ 3 ) e x/ + dx = e e. Answers 3.. () π x 5 x + dx = 5π. 5 x ˆ () π y + 4y dy = π π. 6 (3) (4) (5) ˆ ˆ ˆ 3 πy 3 + 9y 4 dy = π 7 ( ). π6x 37 dx = 6π 37. π (y + ) + 5π dy = (y + ) 3. ( ) ) ( ) dx = 43π

We divide the interval [a, b] into subintervals of equal length x = b a n

We divide the interval [a, b] into subintervals of equal length x = b a n Arc Length Given curve C defined by function f(x), we wnt to find the length of this curve between nd b. We do this by using process similr to wht we did in defining the Riemnn Sum of definite integrl:

More information

a < a+ x < a+2 x < < a+n x = b, n A i n f(x i ) x. i=1 i=1

a < a+ x < a+2 x < < a+n x = b, n A i n f(x i ) x. i=1 i=1 Mth 33 Volume Stewrt 5.2 Geometry of integrls. In this section, we will lern how to compute volumes using integrls defined by slice nlysis. First, we recll from Clculus I how to compute res. Given the

More information

Math 0230 Calculus 2 Lectures

Math 0230 Calculus 2 Lectures Mth Clculus Lectures Chpter 7 Applictions of Integrtion Numertion of sections corresponds to the text Jmes Stewrt, Essentil Clculus, Erly Trnscendentls, Second edition. Section 7. Ares Between Curves Two

More information

5.2 Volumes: Disks and Washers

5.2 Volumes: Disks and Washers 4 pplictions of definite integrls 5. Volumes: Disks nd Wshers In the previous section, we computed volumes of solids for which we could determine the re of cross-section or slice. In this section, we restrict

More information

Not for reproduction

Not for reproduction AREA OF A SURFACE OF REVOLUTION cut h FIGURE FIGURE πr r r l h FIGURE A surfce of revolution is formed when curve is rotted bout line. Such surfce is the lterl boundry of solid of revolution of the type

More information

14.4. Lengths of curves and surfaces of revolution. Introduction. Prerequisites. Learning Outcomes

14.4. Lengths of curves and surfaces of revolution. Introduction. Prerequisites. Learning Outcomes Lengths of curves nd surfces of revolution 4.4 Introduction Integrtion cn be used to find the length of curve nd the re of the surfce generted when curve is rotted round n xis. In this section we stte

More information

l 2 p2 n 4n 2, the total surface area of the

l 2 p2 n 4n 2, the total surface area of the Week 6 Lectures Sections 7.5, 7.6 Section 7.5: Surfce re of Revolution Surfce re of Cone: Let C be circle of rdius r. Let P n be n n-sided regulr polygon of perimeter p n with vertices on C. Form cone

More information

Applications of Definite Integral

Applications of Definite Integral Chpter 5 Applitions of Definite Integrl 5.1 Are Between Two Curves In this setion we use integrls to find res of regions tht lie between the grphs of two funtions. Consider the region tht lies between

More information

Definite integral. Mathematics FRDIS MENDELU

Definite integral. Mathematics FRDIS MENDELU Definite integrl Mthemtics FRDIS MENDELU Simon Fišnrová Brno 1 Motivtion - re under curve Suppose, for simplicity, tht y = f(x) is nonnegtive nd continuous function defined on [, b]. Wht is the re of the

More information

Math 8 Winter 2015 Applications of Integration

Math 8 Winter 2015 Applications of Integration Mth 8 Winter 205 Applictions of Integrtion Here re few importnt pplictions of integrtion. The pplictions you my see on n exm in this course include only the Net Chnge Theorem (which is relly just the Fundmentl

More information

Definite integral. Mathematics FRDIS MENDELU. Simona Fišnarová (Mendel University) Definite integral MENDELU 1 / 30

Definite integral. Mathematics FRDIS MENDELU. Simona Fišnarová (Mendel University) Definite integral MENDELU 1 / 30 Definite integrl Mthemtics FRDIS MENDELU Simon Fišnrová (Mendel University) Definite integrl MENDELU / Motivtion - re under curve Suppose, for simplicity, tht y = f(x) is nonnegtive nd continuous function

More information

Applications of Definite Integral

Applications of Definite Integral Chpter 5 Applitions of Definite Integrl 5.1 Are Between Two Curves In this setion we use integrls to find res of regions tht lie between the grphs of two funtions. Consider the region tht lies between

More information

5 Applications of Definite Integrals

5 Applications of Definite Integrals 5 Applictions of Definite Integrls The previous chpter introduced the concepts of definite integrl s n re nd s limit of Riemnn sums, demonstrted some of the properties of integrls, introduced some methods

More information

Math 113 Exam 1-Review

Math 113 Exam 1-Review Mth 113 Exm 1-Review September 26, 2016 Exm 1 covers 6.1-7.3 in the textbook. It is dvisble to lso review the mteril from 5.3 nd 5.5 s this will be helpful in solving some of the problems. 6.1 Are Between

More information

MAT187H1F Lec0101 Burbulla

MAT187H1F Lec0101 Burbulla Chpter 6 Lecture Notes Review nd Two New Sections Sprint 17 Net Distnce nd Totl Distnce Trvelled Suppose s is the position of prticle t time t for t [, b]. Then v dt = s (t) dt = s(b) s(). s(b) s() is

More information

Chapter 0. What is the Lebesgue integral about?

Chapter 0. What is the Lebesgue integral about? Chpter 0. Wht is the Lebesgue integrl bout? The pln is to hve tutoril sheet ech week, most often on Fridy, (to be done during the clss) where you will try to get used to the ides introduced in the previous

More information

7.1 Integral as Net Change and 7.2 Areas in the Plane Calculus

7.1 Integral as Net Change and 7.2 Areas in the Plane Calculus 7.1 Integrl s Net Chnge nd 7. Ares in the Plne Clculus 7.1 INTEGRAL AS NET CHANGE Notecrds from 7.1: Displcement vs Totl Distnce, Integrl s Net Chnge We hve lredy seen how the position of n oject cn e

More information

A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H. Thomas Shores Department of Mathematics University of Nebraska Spring 2007

A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H. Thomas Shores Department of Mathematics University of Nebraska Spring 2007 A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H Thoms Shores Deprtment of Mthemtics University of Nebrsk Spring 2007 Contents Rtes of Chnge nd Derivtives 1 Dierentils 4 Are nd Integrls 5 Multivrite Clculus

More information

7.6 The Use of Definite Integrals in Physics and Engineering

7.6 The Use of Definite Integrals in Physics and Engineering Arknss Tech University MATH 94: Clculus II Dr. Mrcel B. Finn 7.6 The Use of Definite Integrls in Physics nd Engineering It hs been shown how clculus cn be pplied to find solutions to geometric problems

More information

1 The Riemann Integral

1 The Riemann Integral The Riemnn Integrl. An exmple leding to the notion of integrl (res) We know how to find (i.e. define) the re of rectngle (bse height), tringle ( (sum of res of tringles). But how do we find/define n re

More information

x = a To determine the volume of the solid, we use a definite integral to sum the volumes of the slices as we let!x " 0 :

x = a To determine the volume of the solid, we use a definite integral to sum the volumes of the slices as we let!x  0 : Clculus II MAT 146 Integrtion Applictions: Volumes of 3D Solids Our gol is to determine volumes of vrious shpes. Some of the shpes re the result of rotting curve out n xis nd other shpes re simply given

More information

Review of Calculus, cont d

Review of Calculus, cont d Jim Lmbers MAT 460 Fll Semester 2009-10 Lecture 3 Notes These notes correspond to Section 1.1 in the text. Review of Clculus, cont d Riemnn Sums nd the Definite Integrl There re mny cses in which some

More information

Space Curves. Recall the parametric equations of a curve in xy-plane and compare them with parametric equations of a curve in space.

Space Curves. Recall the parametric equations of a curve in xy-plane and compare them with parametric equations of a curve in space. Clculus 3 Li Vs Spce Curves Recll the prmetric equtions of curve in xy-plne nd compre them with prmetric equtions of curve in spce. Prmetric curve in plne x = x(t) y = y(t) Prmetric curve in spce x = x(t)

More information

MORE FUNCTION GRAPHING; OPTIMIZATION. (Last edited October 28, 2013 at 11:09pm.)

MORE FUNCTION GRAPHING; OPTIMIZATION. (Last edited October 28, 2013 at 11:09pm.) MORE FUNCTION GRAPHING; OPTIMIZATION FRI, OCT 25, 203 (Lst edited October 28, 203 t :09pm.) Exercise. Let n be n rbitrry positive integer. Give n exmple of function with exctly n verticl symptotes. Give

More information

Definite Integrals. The area under a curve can be approximated by adding up the areas of rectangles = 1 1 +

Definite Integrals. The area under a curve can be approximated by adding up the areas of rectangles = 1 1 + Definite Integrls --5 The re under curve cn e pproximted y dding up the res of rectngles. Exmple. Approximte the re under y = from x = to x = using equl suintervls nd + x evluting the function t the left-hnd

More information

Section 6: Area, Volume, and Average Value

Section 6: Area, Volume, and Average Value Chpter The Integrl Applied Clculus Section 6: Are, Volume, nd Averge Vlue Are We hve lredy used integrls to find the re etween the grph of function nd the horizontl xis. Integrls cn lso e used to find

More information

Math 120 Answers for Homework 13

Math 120 Answers for Homework 13 Mth 12 Answers for Homework 13 1. In this problem we will use the fct tht if m f(x M on n intervl [, b] (nd if f is integrble on [, b] then (* m(b f dx M(b. ( The function f(x = 1 + x 3 is n incresing

More information

Line Integrals. Partitioning the Curve. Estimating the Mass

Line Integrals. Partitioning the Curve. Estimating the Mass Line Integrls Suppose we hve curve in the xy plne nd ssocite density δ(p ) = δ(x, y) t ech point on the curve. urves, of course, do not hve density or mss, but it my sometimes be convenient or useful to

More information

Test , 8.2, 8.4 (density only), 8.5 (work only), 9.1, 9.2 and 9.3 related test 1 material and material from prior classes

Test , 8.2, 8.4 (density only), 8.5 (work only), 9.1, 9.2 and 9.3 related test 1 material and material from prior classes Test 2 8., 8.2, 8.4 (density only), 8.5 (work only), 9., 9.2 nd 9.3 relted test mteril nd mteril from prior clsses Locl to Globl Perspectives Anlyze smll pieces to understnd the big picture. Exmples: numericl

More information

MTH 122 Fall 2008 Essex County College Division of Mathematics Handout Version 10 1 October 14, 2008

MTH 122 Fall 2008 Essex County College Division of Mathematics Handout Version 10 1 October 14, 2008 MTH 22 Fll 28 Essex County College Division of Mthemtics Hndout Version October 4, 28 Arc Length Everyone should be fmilir with the distnce formul tht ws introduced in elementry lgebr. It is bsic formul

More information

APPLICATIONS OF DEFINITE INTEGRALS

APPLICATIONS OF DEFINITE INTEGRALS Chpter 6 APPICATIONS OF DEFINITE INTEGRAS OVERVIEW In Chpter 5 we discovered the connection etween Riemnn sums ssocited with prtition P of the finite closed intervl [, ] nd the process of integrtion. We

More information

Math 113 Exam 2 Practice

Math 113 Exam 2 Practice Mth 3 Exm Prctice Februry 8, 03 Exm will cover 7.4, 7.5, 7.7, 7.8, 8.-3 nd 8.5. Plese note tht integrtion skills lerned in erlier sections will still be needed for the mteril in 7.5, 7.8 nd chpter 8. This

More information

Sections 5.2: The Definite Integral

Sections 5.2: The Definite Integral Sections 5.2: The Definite Integrl In this section we shll formlize the ides from the lst section to functions in generl. We strt with forml definition.. The Definite Integrl Definition.. Suppose f(x)

More information

Math 107H Topics for the first exam. csc 2 x dx = cot x + C csc x cotx dx = csc x + C tan x dx = ln secx + C cot x dx = ln sinx + C e x dx = e x + C

Math 107H Topics for the first exam. csc 2 x dx = cot x + C csc x cotx dx = csc x + C tan x dx = ln secx + C cot x dx = ln sinx + C e x dx = e x + C Integrtion Mth 07H Topics for the first exm Bsic list: x n dx = xn+ + C (provided n ) n + sin(kx) dx = cos(kx) + C k sec x dx = tnx + C sec x tnx dx = sec x + C /x dx = ln x + C cos(kx) dx = sin(kx) +

More information

[ ( ) ( )] Section 6.1 Area of Regions between two Curves. Goals: 1. To find the area between two curves

[ ( ) ( )] Section 6.1 Area of Regions between two Curves. Goals: 1. To find the area between two curves Gols: 1. To find the re etween two curves Section 6.1 Are of Regions etween two Curves I. Are of Region Between Two Curves A. Grphicl Represention = _ B. Integrl Represention [ ( ) ( )] f x g x dx = C.

More information

Module M5.4 Applications of integration

Module M5.4 Applications of integration F L E X I B L E L E A R N I N G A P P R O A C H T O P H Y S I C S Module M5.4 Opening items. Module introduction. Fst trck questions. Redy to study? Ares. Are under grph. Are between two grphs Solids of

More information

k ) and directrix x = h p is A focal chord is a line segment which passes through the focus of a parabola and has endpoints on the parabola.

k ) and directrix x = h p is A focal chord is a line segment which passes through the focus of a parabola and has endpoints on the parabola. Stndrd Eqution of Prol with vertex ( h, k ) nd directrix y = k p is ( x h) p ( y k ) = 4. Verticl xis of symmetry Stndrd Eqution of Prol with vertex ( h, k ) nd directrix x = h p is ( y k ) p( x h) = 4.

More information

JUST THE MATHS UNIT NUMBER INTEGRATION APPLICATIONS 8 (First moments of a volume) A.J.Hobson

JUST THE MATHS UNIT NUMBER INTEGRATION APPLICATIONS 8 (First moments of a volume) A.J.Hobson JUST THE MATHS UNIT NUMBER 3.8 INTEGRATIN APPLICATINS 8 (First moments of volume) b A.J.Hobson 3.8. Introduction 3.8. First moment of volume of revolution bout plne through the origin, perpendiculr to

More information

Integration Techniques

Integration Techniques Integrtion Techniques. Integrtion of Trigonometric Functions Exmple. Evlute cos x. Recll tht cos x = cos x. Hence, cos x Exmple. Evlute = ( + cos x) = (x + sin x) + C = x + 4 sin x + C. cos 3 x. Let u

More information

The Riemann Integral

The Riemann Integral Deprtment of Mthemtics King Sud University 2017-2018 Tble of contents 1 Anti-derivtive Function nd Indefinite Integrls 2 3 4 5 Indefinite Integrls & Anti-derivtive Function Definition Let f : I R be function

More information

Test 3 Review. Jiwen He. I will replace your lowest test score with the percentage grade from the final exam (provided it is higher).

Test 3 Review. Jiwen He. I will replace your lowest test score with the percentage grade from the final exam (provided it is higher). Test 3 Review Jiwen He Test 3 Test 3: Dec. 4-6 in CASA Mteril - Through 6.3. No Homework (Thnksgiving) No homework this week! Hve GREAT Thnksgiving! Finl Exm Finl Exm: Dec. 14-17 in CASA You Might Be Interested

More information

Geometric and Mechanical Applications of Integrals

Geometric and Mechanical Applications of Integrals 5 Geometric nd Mechnicl Applictions of Integrls 5.1 Computing Are 5.1.1 Using Crtesin Coordintes Suppose curve is given by n eqution y = f(x), x b, where f : [, b] R is continuous function such tht f(x)

More information

Sample Problems for the Final of Math 121, Fall, 2005

Sample Problems for the Final of Math 121, Fall, 2005 Smple Problems for the Finl of Mth, Fll, 5 The following is collection of vrious types of smple problems covering sections.8,.,.5, nd.8 6.5 of the text which constitute only prt of the common Mth Finl.

More information

Indefinite Integral. Chapter Integration - reverse of differentiation

Indefinite Integral. Chapter Integration - reverse of differentiation Chpter Indefinite Integrl Most of the mthemticl opertions hve inverse opertions. The inverse opertion of differentition is clled integrtion. For exmple, describing process t the given moment knowing the

More information

7.1 Integral as Net Change Calculus. What is the total distance traveled? What is the total displacement?

7.1 Integral as Net Change Calculus. What is the total distance traveled? What is the total displacement? 7.1 Integrl s Net Chnge Clculus 7.1 INTEGRAL AS NET CHANGE Distnce versus Displcement We hve lredy seen how the position of n oject cn e found y finding the integrl of the velocity function. The chnge

More information

Definition of Continuity: The function f(x) is continuous at x = a if f(a) exists and lim

Definition of Continuity: The function f(x) is continuous at x = a if f(a) exists and lim Mth 9 Course Summry/Study Guide Fll, 2005 [1] Limits Definition of Limit: We sy tht L is the limit of f(x) s x pproches if f(x) gets closer nd closer to L s x gets closer nd closer to. We write lim f(x)

More information

4.4 Areas, Integrals and Antiderivatives

4.4 Areas, Integrals and Antiderivatives . res, integrls nd ntiderivtives 333. Ares, Integrls nd Antiderivtives This section explores properties of functions defined s res nd exmines some connections mong res, integrls nd ntiderivtives. In order

More information

The Fundamental Theorem of Calculus. The Total Change Theorem and the Area Under a Curve.

The Fundamental Theorem of Calculus. The Total Change Theorem and the Area Under a Curve. Clculus Li Vs The Fundmentl Theorem of Clculus. The Totl Chnge Theorem nd the Are Under Curve. Recll the following fct from Clculus course. If continuous function f(x) represents the rte of chnge of F

More information

APPM 1360 Exam 2 Spring 2016

APPM 1360 Exam 2 Spring 2016 APPM 6 Em Spring 6. 8 pts, 7 pts ech For ech of the following prts, let f + nd g 4. For prts, b, nd c, set up, but do not evlute, the integrl needed to find the requested informtion. The volume of the

More information

Math 113 Fall Final Exam Review. 2. Applications of Integration Chapter 6 including sections and section 6.8

Math 113 Fall Final Exam Review. 2. Applications of Integration Chapter 6 including sections and section 6.8 Mth 3 Fll 0 The scope of the finl exm will include: Finl Exm Review. Integrls Chpter 5 including sections 5. 5.7, 5.0. Applictions of Integrtion Chpter 6 including sections 6. 6.5 nd section 6.8 3. Infinite

More information

Riemann Sums and Riemann Integrals

Riemann Sums and Riemann Integrals Riemnn Sums nd Riemnn Integrls Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University August 26, 2013 Outline 1 Riemnn Sums 2 Riemnn Integrls 3 Properties

More information

Math& 152 Section Integration by Parts

Math& 152 Section Integration by Parts Mth& 5 Section 7. - Integrtion by Prts Integrtion by prts is rule tht trnsforms the integrl of the product of two functions into other (idelly simpler) integrls. Recll from Clculus I tht given two differentible

More information

Further integration. x n nx n 1 sinh x cosh x log x 1/x cosh x sinh x e x e x tan x sec 2 x sin x cos x tan 1 x 1/(1 + x 2 ) cos x sin x

Further integration. x n nx n 1 sinh x cosh x log x 1/x cosh x sinh x e x e x tan x sec 2 x sin x cos x tan 1 x 1/(1 + x 2 ) cos x sin x Further integrtion Stndrd derivtives nd integrls The following cn be thought of s list of derivtives or eqully (red bckwrds) s list of integrls. Mke sure you know them! There ren t very mny. f(x) f (x)

More information

Riemann Sums and Riemann Integrals

Riemann Sums and Riemann Integrals Riemnn Sums nd Riemnn Integrls Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University August 26, 203 Outline Riemnn Sums Riemnn Integrls Properties Abstrct

More information

INTRODUCTION TO INTEGRATION

INTRODUCTION TO INTEGRATION INTRODUCTION TO INTEGRATION 5.1 Ares nd Distnces Assume f(x) 0 on the intervl [, b]. Let A be the re under the grph of f(x). b We will obtin n pproximtion of A in the following three steps. STEP 1: Divide

More information

The Wave Equation I. MA 436 Kurt Bryan

The Wave Equation I. MA 436 Kurt Bryan 1 Introduction The Wve Eqution I MA 436 Kurt Bryn Consider string stretching long the x xis, of indeterminte (or even infinite!) length. We wnt to derive n eqution which models the motion of the string

More information

Line Integrals. Chapter Definition

Line Integrals. Chapter Definition hpter 2 Line Integrls 2.1 Definition When we re integrting function of one vrible, we integrte long n intervl on one of the xes. We now generlize this ide by integrting long ny curve in the xy-plne. It

More information

Partial Derivatives. Limits. For a single variable function f (x), the limit lim

Partial Derivatives. Limits. For a single variable function f (x), the limit lim Limits Prtil Derivtives For single vrible function f (x), the limit lim x f (x) exists only if the right-hnd side limit equls to the left-hnd side limit, i.e., lim f (x) = lim f (x). x x + For two vribles

More information

dt. However, we might also be curious about dy

dt. However, we might also be curious about dy Section 0. The Clculus of Prmetric Curves Even though curve defined prmetricly my not be function, we cn still consider concepts such s rtes of chnge. However, the concepts will need specil tretment. For

More information

Mathematics. Area under Curve.

Mathematics. Area under Curve. Mthemtics Are under Curve www.testprepkrt.com Tle of Content 1. Introduction.. Procedure of Curve Sketching. 3. Sketching of Some common Curves. 4. Are of Bounded Regions. 5. Sign convention for finding

More information

The Regulated and Riemann Integrals

The Regulated and Riemann Integrals Chpter 1 The Regulted nd Riemnn Integrls 1.1 Introduction We will consider severl different pproches to defining the definite integrl f(x) dx of function f(x). These definitions will ll ssign the sme vlue

More information

Exam 2, Mathematics 4701, Section ETY6 6:05 pm 7:40 pm, March 31, 2016, IH-1105 Instructor: Attila Máté 1

Exam 2, Mathematics 4701, Section ETY6 6:05 pm 7:40 pm, March 31, 2016, IH-1105 Instructor: Attila Máté 1 Exm, Mthemtics 471, Section ETY6 6:5 pm 7:4 pm, Mrch 1, 16, IH-115 Instructor: Attil Máté 1 17 copies 1. ) Stte the usul sufficient condition for the fixed-point itertion to converge when solving the eqution

More information

AP Calculus BC Review Applications of Integration (Chapter 6) noting that one common instance of a force is weight

AP Calculus BC Review Applications of Integration (Chapter 6) noting that one common instance of a force is weight AP Clculus BC Review Applictions of Integrtion (Chpter Things to Know n Be Able to Do Fin the re between two curves by integrting with respect to x or y Fin volumes by pproximtions with cross sections:

More information

5.7 Improper Integrals

5.7 Improper Integrals 458 pplictions of definite integrls 5.7 Improper Integrls In Section 5.4, we computed the work required to lift pylod of mss m from the surfce of moon of mss nd rdius R to height H bove the surfce of the

More information

Notes on Calculus II Integral Calculus. Miguel A. Lerma

Notes on Calculus II Integral Calculus. Miguel A. Lerma Notes on Clculus II Integrl Clculus Miguel A. Lerm November 22, 22 Contents Introduction 5 Chpter. Integrls 6.. Ares nd Distnces. The Definite Integrl 6.2. The Evlution Theorem.3. The Fundmentl Theorem

More information

Section 4.8. D v(t j 1 ) t. (4.8.1) j=1

Section 4.8. D v(t j 1 ) t. (4.8.1) j=1 Difference Equtions to Differentil Equtions Section.8 Distnce, Position, nd the Length of Curves Although we motivted the definition of the definite integrl with the notion of re, there re mny pplictions

More information

We partition C into n small arcs by forming a partition of [a, b] by picking s i as follows: a = s 0 < s 1 < < s n = b.

We partition C into n small arcs by forming a partition of [a, b] by picking s i as follows: a = s 0 < s 1 < < s n = b. Mth 255 - Vector lculus II Notes 4.2 Pth nd Line Integrls We begin with discussion of pth integrls (the book clls them sclr line integrls). We will do this for function of two vribles, but these ides cn

More information

12 TRANSFORMING BIVARIATE DENSITY FUNCTIONS

12 TRANSFORMING BIVARIATE DENSITY FUNCTIONS 1 TRANSFORMING BIVARIATE DENSITY FUNCTIONS Hving seen how to trnsform the probbility density functions ssocited with single rndom vrible, the next logicl step is to see how to trnsform bivrite probbility

More information

Riemann Integrals and the Fundamental Theorem of Calculus

Riemann Integrals and the Fundamental Theorem of Calculus Riemnn Integrls nd the Fundmentl Theorem of Clculus Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University September 16, 2013 Outline Grphing Riemnn Sums

More information

NUMERICAL INTEGRATION. The inverse process to differentiation in calculus is integration. Mathematically, integration is represented by.

NUMERICAL INTEGRATION. The inverse process to differentiation in calculus is integration. Mathematically, integration is represented by. NUMERICAL INTEGRATION 1 Introduction The inverse process to differentition in clculus is integrtion. Mthemticlly, integrtion is represented by f(x) dx which stnds for the integrl of the function f(x) with

More information

Properties of Integrals, Indefinite Integrals. Goals: Definition of the Definite Integral Integral Calculations using Antiderivatives

Properties of Integrals, Indefinite Integrals. Goals: Definition of the Definite Integral Integral Calculations using Antiderivatives Block #6: Properties of Integrls, Indefinite Integrls Gols: Definition of the Definite Integrl Integrl Clcultions using Antiderivtives Properties of Integrls The Indefinite Integrl 1 Riemnn Sums - 1 Riemnn

More information

Thammasat University Department of Common and Graduate Studies

Thammasat University Department of Common and Graduate Studies Sirindhorn Interntionl Institute of Technology Thmmst University Deprtment of Common nd Grdute Studies Semester: 3/2008 Instructors: Dr. Prpun Suksompong MAS 6: Lecture Notes 7 6 Applictions of the Definite

More information

Polynomials and Division Theory

Polynomials and Division Theory Higher Checklist (Unit ) Higher Checklist (Unit ) Polynomils nd Division Theory Skill Achieved? Know tht polynomil (expression) is of the form: n x + n x n + n x n + + n x + x + 0 where the i R re the

More information

Physics 116C Solution of inhomogeneous ordinary differential equations using Green s functions

Physics 116C Solution of inhomogeneous ordinary differential equations using Green s functions Physics 6C Solution of inhomogeneous ordinry differentil equtions using Green s functions Peter Young November 5, 29 Homogeneous Equtions We hve studied, especilly in long HW problem, second order liner

More information

Math 426: Probability Final Exam Practice

Math 426: Probability Final Exam Practice Mth 46: Probbility Finl Exm Prctice. Computtionl problems 4. Let T k (n) denote the number of prtitions of the set {,..., n} into k nonempty subsets, where k n. Argue tht T k (n) kt k (n ) + T k (n ) by

More information

f(x) dx, If one of these two conditions is not met, we call the integral improper. Our usual definition for the value for the definite integral

f(x) dx, If one of these two conditions is not met, we call the integral improper. Our usual definition for the value for the definite integral Improper Integrls Every time tht we hve evluted definite integrl such s f(x) dx, we hve mde two implicit ssumptions bout the integrl:. The intervl [, b] is finite, nd. f(x) is continuous on [, b]. If one

More information

THE EXISTENCE-UNIQUENESS THEOREM FOR FIRST-ORDER DIFFERENTIAL EQUATIONS.

THE EXISTENCE-UNIQUENESS THEOREM FOR FIRST-ORDER DIFFERENTIAL EQUATIONS. THE EXISTENCE-UNIQUENESS THEOREM FOR FIRST-ORDER DIFFERENTIAL EQUATIONS RADON ROSBOROUGH https://intuitiveexplntionscom/picrd-lindelof-theorem/ This document is proof of the existence-uniqueness theorem

More information

The Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus The Fundmentl Theorem of Clculus MATH 151 Clculus for Mngement J. Robert Buchnn Deprtment of Mthemtics Fll 2018 Objectives Define nd evlute definite integrls using the concept of re. Evlute definite integrls

More information

Goals: Determine how to calculate the area described by a function. Define the definite integral. Explore the relationship between the definite

Goals: Determine how to calculate the area described by a function. Define the definite integral. Explore the relationship between the definite Unit #8 : The Integrl Gols: Determine how to clculte the re described by function. Define the definite integrl. Eplore the reltionship between the definite integrl nd re. Eplore wys to estimte the definite

More information

Conducting Ellipsoid and Circular Disk

Conducting Ellipsoid and Circular Disk 1 Problem Conducting Ellipsoid nd Circulr Disk Kirk T. McDonld Joseph Henry Lbortories, Princeton University, Princeton, NJ 08544 (September 1, 00) Show tht the surfce chrge density σ on conducting ellipsoid,

More information

Chapter 6 Notes, Larson/Hostetler 3e

Chapter 6 Notes, Larson/Hostetler 3e Contents 6. Antiderivtives nd the Rules of Integrtion.......................... 6. Are nd the Definite Integrl.................................. 6.. Are............................................ 6. Reimnn

More information

Advanced Calculus: MATH 410 Notes on Integrals and Integrability Professor David Levermore 17 October 2004

Advanced Calculus: MATH 410 Notes on Integrals and Integrability Professor David Levermore 17 October 2004 Advnced Clculus: MATH 410 Notes on Integrls nd Integrbility Professor Dvid Levermore 17 October 2004 1. Definite Integrls In this section we revisit the definite integrl tht you were introduced to when

More information

Riemann is the Mann! (But Lebesgue may besgue to differ.)

Riemann is the Mann! (But Lebesgue may besgue to differ.) Riemnn is the Mnn! (But Lebesgue my besgue to differ.) Leo Livshits My 2, 2008 1 For finite intervls in R We hve seen in clss tht every continuous function f : [, b] R hs the property tht for every ɛ >

More information

x 2 1 dx x 3 dx = ln(x) + 2e u du = 2e u + C = 2e x + C 2x dx = arcsin x + 1 x 1 x du = 2 u + C (t + 2) 50 dt x 2 4 dx

x 2 1 dx x 3 dx = ln(x) + 2e u du = 2e u + C = 2e x + C 2x dx = arcsin x + 1 x 1 x du = 2 u + C (t + 2) 50 dt x 2 4 dx . Compute the following indefinite integrls: ) sin(5 + )d b) c) d e d d) + d Solutions: ) After substituting u 5 +, we get: sin(5 + )d sin(u)du cos(u) + C cos(5 + ) + C b) We hve: d d ln() + + C c) Substitute

More information

1 The fundamental theorems of calculus.

1 The fundamental theorems of calculus. The fundmentl theorems of clculus. The fundmentl theorems of clculus. Evluting definite integrls. The indefinite integrl- new nme for nti-derivtive. Differentiting integrls. Tody we provide the connection

More information

Section 14.3 Arc Length and Curvature

Section 14.3 Arc Length and Curvature Section 4.3 Arc Length nd Curvture Clculus on Curves in Spce In this section, we ly the foundtions for describing the movement of n object in spce.. Vector Function Bsics In Clc, formul for rc length in

More information

7.2 Riemann Integrable Functions

7.2 Riemann Integrable Functions 7.2 Riemnn Integrble Functions Theorem 1. If f : [, b] R is step function, then f R[, b]. Theorem 2. If f : [, b] R is continuous on [, b], then f R[, b]. Theorem 3. If f : [, b] R is bounded nd continuous

More information

10. AREAS BETWEEN CURVES

10. AREAS BETWEEN CURVES . AREAS BETWEEN CURVES.. Ares etween curves So res ove the x-xis re positive nd res elow re negtive, right? Wrong! We lied! Well, when you first lern out integrtion it s convenient fiction tht s true in

More information

Review of basic calculus

Review of basic calculus Review of bsic clculus This brief review reclls some of the most importnt concepts, definitions, nd theorems from bsic clculus. It is not intended to tech bsic clculus from scrtch. If ny of the items below

More information

CHAPTER 4 MULTIPLE INTEGRALS

CHAPTER 4 MULTIPLE INTEGRALS CHAPTE 4 MULTIPLE INTEGAL The objects of this chpter re five-fold. They re: (1 Discuss when sclr-vlued functions f cn be integrted over closed rectngulr boxes in n ; simply put, f is integrble over iff

More information

Main topics for the First Midterm

Main topics for the First Midterm Min topics for the First Midterm The Midterm will cover Section 1.8, Chpters 2-3, Sections 4.1-4.8, nd Sections 5.1-5.3 (essentilly ll of the mteril covered in clss). Be sure to know the results of the

More information

1 Part II: Numerical Integration

1 Part II: Numerical Integration Mth 4 Lb 1 Prt II: Numericl Integrtion This section includes severl techniques for getting pproimte numericl vlues for definite integrls without using ntiderivtives. Mthemticll, ect nswers re preferble

More information

Calculus of Variations

Calculus of Variations Clculus of Vritions Com S 477/577 Notes) Yn-Bin Ji Dec 4, 2017 1 Introduction A functionl ssigns rel number to ech function or curve) in some clss. One might sy tht functionl is function of nother function

More information

APPLICATIONS OF THE DEFINITE INTEGRAL IN GEOMETRY, SCIENCE, AND ENGINEERING

APPLICATIONS OF THE DEFINITE INTEGRAL IN GEOMETRY, SCIENCE, AND ENGINEERING 6 Courtes NASA APPLICATIONS OF THE DEFINITE INTEGRAL IN GEOMETRY, SCIENCE, AND ENGINEERING Clculus is essentil for the computtions required to lnd n stronut on the Moon. In the lst chpter we introduced

More information

Homework 4. (1) If f R[a, b], show that f 3 R[a, b]. If f + (x) = max{f(x), 0}, is f + R[a, b]? Justify your answer.

Homework 4. (1) If f R[a, b], show that f 3 R[a, b]. If f + (x) = max{f(x), 0}, is f + R[a, b]? Justify your answer. Homework 4 (1) If f R[, b], show tht f 3 R[, b]. If f + (x) = mx{f(x), 0}, is f + R[, b]? Justify your nswer. (2) Let f be continuous function on [, b] tht is strictly positive except finitely mny points

More information

1. Find the derivative of the following functions. a) f(x) = 2 + 3x b) f(x) = (5 2x) 8 c) f(x) = e2x

1. Find the derivative of the following functions. a) f(x) = 2 + 3x b) f(x) = (5 2x) 8 c) f(x) = e2x I. Dierentition. ) Rules. *product rule, quotient rule, chin rule MATH 34B FINAL REVIEW. Find the derivtive of the following functions. ) f(x) = 2 + 3x x 3 b) f(x) = (5 2x) 8 c) f(x) = e2x 4x 7 +x+2 d)

More information

Practice Final. Name: Problem 1. Show all of your work, label your answers clearly, and do not use a calculator.

Practice Final. Name: Problem 1. Show all of your work, label your answers clearly, and do not use a calculator. Nme: MATH 2250 Clculus Eric Perkerson Dte: December 11, 2015 Prctice Finl Show ll of your work, lbel your nswers clerly, nd do not use clcultor. Problem 1 Compute the following limits, showing pproprite

More information

Homework Assignment 6 Solution Set

Homework Assignment 6 Solution Set Homework Assignment 6 Solution Set PHYCS 440 Mrch, 004 Prolem (Griffiths 4.6 One wy to find the energy is to find the E nd D fields everywhere nd then integrte the energy density for those fields. We know

More information

Improper Integrals. Type I Improper Integrals How do we evaluate an integral such as

Improper Integrals. Type I Improper Integrals How do we evaluate an integral such as Improper Integrls Two different types of integrls cn qulify s improper. The first type of improper integrl (which we will refer to s Type I) involves evluting n integrl over n infinite region. In the grph

More information

In Mathematics for Construction, we learnt that

In Mathematics for Construction, we learnt that III DOUBLE INTEGATION THE ANTIDEIVATIVE OF FUNCTIONS OF VAIABLES In Mthemtics or Construction, we lernt tht the indeinite integrl is the ntiderivtive o ( d ( Double Integrtion Pge Hence d d ( d ( The ntiderivtive

More information