( ) ( ) Math 17 Exam II Solutions
|
|
- Alison Gardner
- 5 years ago
- Views:
Transcription
1 Math 7 Exam II Solutions. Sketch the vector field F(x,y) -yi + xj by drawing a few vectors. Draw the vectors associated with at least one point in each quadrant and draw the vectors associated with at least one point on the positive x axis, one on the negative x axis, one on the positive y axis, and one on the negative y axis.. G is the solid in the first octant bounded by the coordinate planes and the plane x+y+z. The density of G is given by δ(x,y,z) x y 6 +z. Set up, but do not evaluate, an iterated triple integral to compute the mass of G. x y M ( x) z x y ( x y 6 + z )dzdydx x0 z0 Other orders of integration are possible.. Show that the volume of a sphere of radius a is πa. Cylindrical coordinates: z a r rdzdrdθ z0 z a rz r z0 drdθ r a r drdθ ra a r dθ a dθ a θ a ( ) π πa
2 Spherical Coordinates: φπ ρa ( ρ sin φ)dρdφdθ φ0 ρ0 ρa dφdθ φπ ρ sinφ φ0 ρ0 φπ a sinφ φ0 dφdθ a a a φπ sinφ dφdθ φ0 cosφ φ π dθ φ0 ( +)dθ a θ a ( )( π ) πa. Find the centroid of the solid bounded by the cone z x + y and the plane z. (Hint: Symmetry will be helpful.) We first observe that, by symmetry considerations, x0 and y 0. We must find z. We shall use cylindrical coordinates. Note that the intersection of the cone z x + y and the plane z is the circle x + y. We first compute the volume, which is the denominator of z. r z rdzdrdθ z r r rz zr z drdθ r ( r r )drdθ r r r ( ) dθ r 0 7 ( 9 )dθ 9 ( )θ
3 Then, z r z rzdzdrdθ z r r rz z drdθ z r r 9r ( r )drdθ 9r ( r r 8 ) dθ r 0 8 ( 8 8 )dθ ( 8 )θ ( ) 8 π 9 8π 9 Hence, the centroid is ( 0,0, 9 ). 5. An object moves from the point (-5,0) to the point (0,5) and is acted on by the force field F(x,y) 5 xyi + y j. Compute the work done by the force field on the object if the object moves a. along the straight-line segment from (-5,0) to (0,5). The vector-valued function that describes this motion is: r(t) <-5,0> + t<5,5> <-5+5t,5t> t 0 Then, F(r(t)) <-5t+5t,5t > and r'(t) <5,5>. Thus W t (F(r(t)) (r'(t))dt t0 t ( 5t + 5t,5t 5,5 ) dt t0 t ( 5t + 5t +5t )dt t0 t (50t 5t)dt t0 50t 5t t ( ) t
4 b. along the circle with center at the origin and radius 5, counterclockwise from (-5,0) to (0,5). (So, the object moves three quarters of the way around this circle.) The vector-valued function that describes this motion is: r(t) <5cost,5sint> Then, t 5π π F(r(t)) 5 (5sin t cost),5sin t 5sin t cos t,5sin t and r'(t) <-5sint,5cost>. Thus W t 5π (F(r(t)) (r'(t ))dt t π t ( 5sin t cos t,5sin t 5sin t,5cos t ) 5π dt t π t 5π ( 5sin t cost +5sin t cost )dt t π t 5π 00sin t cost dt t π 00sin t 00 t 5π tπ x 0 z 8 x y 6. Consider the integral x dzdy x dx y x z x +y. Find an equivalent integral in cylindrical coordinates. Do not evaluate this integral. Your answer should include the integrand and all limits of integration. The middle and outside limits of integration tell us that the relevant region in the xy plane is the region shown. Converting z x +y to cylindrical coordinates yields zr and converting z 8-x -y to cylindrical coordinates yields z 8-r. Hence, x0 z8 x y z8 r x x dzdy dx y x zx +y r cosθdzdrdθ θπ z r θ π r
5 7. Find the mass of the solid in the first octant that is inside the sphere x +y +z and outside the cone z x + y if its density is given by δ(x,y,z) x + y + z. We shall use spherical coordinates. δ x + y + z ρ. x +y +z is a sphere with center at the origin and radius. z x + y is a cone with vertex at the origin, opening upward, making an angle of π with the positive z axis. Since we are restricting our solid to the first octant, θ goes from 0 to π. Thus, M θ π φ π ρ ρ( ρ sinφ)dρdφdθ θ π φ π φ π ρ0 ρ ρ sin φdρdφdθ θ π φ π φ π ρ0 ρ dφdθ ρ sinφ φ π ρ0 θ π φ π sinφ dφdθ θ π φ π φ π sinφ dφdθ θ π φ π φ cosφ π φ dθ π θ π dθ θ θ π ( ) π π π
Archive of Calculus IV Questions Noel Brady Department of Mathematics University of Oklahoma
Archive of Calculus IV Questions Noel Brady Department of Mathematics University of Oklahoma This is an archive of past Calculus IV exam questions. You should first attempt the questions without looking
More information(b) Find the range of h(x, y) (5) Use the definition of continuity to explain whether or not the function f(x, y) is continuous at (0, 0)
eview Exam Math 43 Name Id ead each question carefully. Avoid simple mistakes. Put a box around the final answer to a question (use the back of the page if necessary). For full credit you must show your
More informationNote: Each problem is worth 14 points except numbers 5 and 6 which are 15 points. = 3 2
Math Prelim II Solutions Spring Note: Each problem is worth points except numbers 5 and 6 which are 5 points. x. Compute x da where is the region in the second quadrant between the + y circles x + y and
More informationInstructions: No books. No notes. Non-graphing calculators only. You are encouraged, although not required, to show your work.
Exam 3 Math 850-007 Fall 04 Odenthal Name: Instructions: No books. No notes. Non-graphing calculators only. You are encouraged, although not required, to show your work.. Evaluate the iterated integral
More informationTufts University Math 13 Department of Mathematics April 2, 2012 Exam 2 12:00 pm to 1:20 pm
Tufts University Math Department of Mathematics April, Eam : pm to : pm Instructions: No calculators, notes or books are allowed. Unless otherwise stated, you must show all work to receive full credit.
More informationMath 6A Practice Problems II
Math 6A Practice Problems II Written by Victoria Kala vtkala@math.ucsb.edu SH 64u Office Hours: R : :pm Last updated 5//6 Answers This page contains answers only. Detailed solutions are on the following
More informationMath Review for Exam Compute the second degree Taylor polynomials about (0, 0) of the following functions: (a) f(x, y) = e 2x 3y.
Math 35 - Review for Exam 1. Compute the second degree Taylor polynomial of f e x+3y about (, ). Solution. A computation shows that f x(, ), f y(, ) 3, f xx(, ) 4, f yy(, ) 9, f xy(, ) 6. The second degree
More informationMultiple Choice. Compute the Jacobian, (u, v), of the coordinate transformation x = u2 v 4, y = uv. (a) 2u 2 + 4v 4 (b) xu yv (c) 3u 2 + 7v 6
.(5pts) y = uv. ompute the Jacobian, Multiple hoice (x, y) (u, v), of the coordinate transformation x = u v 4, (a) u + 4v 4 (b) xu yv (c) u + 7v 6 (d) u (e) u v uv 4 Solution. u v 4v u = u + 4v 4..(5pts)
More informationMath 233. Practice Problems Chapter 15. i j k
Math 233. Practice Problems hapter 15 1. ompute the curl and divergence of the vector field F given by F (4 cos(x 2 ) 2y)i + (4 sin(y 2 ) + 6x)j + (6x 2 y 6x + 4e 3z )k olution: The curl of F is computed
More informationMTH 234 Solutions to Exam 2 April 13, Multiple Choice. Circle the best answer. No work needed. No partial credit available.
MTH 234 Solutions to Exam 2 April 3, 25 Multiple Choice. Circle the best answer. No work needed. No partial credit available.. (5 points) Parametrize of the part of the plane 3x+2y +z = that lies above
More informatione x3 dx dy. 0 y x 2, 0 x 1.
Problem 1. Evaluate by changing the order of integration y e x3 dx dy. Solution:We change the order of integration over the region y x 1. We find and x e x3 dy dx = y x, x 1. x e x3 dx = 1 x=1 3 ex3 x=
More informationAssignment 11 Solutions
. Evaluate Math 9 Assignment olutions F n d, where F bxy,bx y,(x + y z and is the closed surface bounding the region consisting of the solid cylinder x + y a and z b. olution This is a problem for which
More informationVector Calculus. Dr. D. Sukumar
Vector Calculus Dr. D. Sukumar Space co-ordinates Change of variable Cartesian co-ordinates < x < Cartesian co-ordinates < x < < y < Cartesian co-ordinates < x < < y < < z < Cylindrical Cylindrical Cylindrical
More informationPractice Problems for Exam 3 (Solutions) 1. Let F(x, y) = xyi+(y 3x)j, and let C be the curve r(t) = ti+(3t t 2 )j for 0 t 2. Compute F dr.
1. Let F(x, y) xyi+(y 3x)j, and let be the curve r(t) ti+(3t t 2 )j for t 2. ompute F dr. Solution. F dr b a 2 2 F(r(t)) r (t) dt t(3t t 2 ), 3t t 2 3t 1, 3 2t dt t 3 dt 1 2 4 t4 4. 2. Evaluate the line
More informationIntegrals in cylindrical, spherical coordinates (Sect. 15.7)
Integrals in clindrical, spherical coordinates (Sect. 15.7 Integration in spherical coordinates. Review: Clindrical coordinates. Spherical coordinates in space. Triple integral in spherical coordinates.
More informationMcGill University April Calculus 3. Tuesday April 29, 2014 Solutions
McGill University April 4 Faculty of Science Final Examination Calculus 3 Math Tuesday April 9, 4 Solutions Problem (6 points) Let r(t) = (t, cos t, sin t). i. Find the velocity r (t) and the acceleration
More information1. Find and classify the extrema of h(x, y) = sin(x) sin(y) sin(x + y) on the square[0, π] [0, π]. (Keep in mind there is a boundary to check out).
. Find and classify the extrema of hx, y sinx siny sinx + y on the square[, π] [, π]. Keep in mind there is a boundary to check out. Solution: h x cos x sin y sinx + y + sin x sin y cosx + y h y sin x
More informationMATH 52 FINAL EXAM SOLUTIONS
MAH 5 FINAL EXAM OLUION. (a) ketch the region R of integration in the following double integral. x xe y5 dy dx R = {(x, y) x, x y }. (b) Express the region R as an x-simple region. R = {(x, y) y, x y }
More informationDimensions = xyz dv. xyz dv as an iterated integral in rectangular coordinates.
Math Show Your Work! Page of 8. () A rectangular box is to hold 6 cubic meters. The material used for the top and bottom of the box is twice as expensive per square meter than the material used for the
More informationIn general, the formula is S f ds = D f(φ(u, v)) Φ u Φ v da. To compute surface area, we choose f = 1. We compute
alculus III Test 3 ample Problem Answers/olutions 1. Express the area of the surface Φ(u, v) u cosv, u sinv, 2v, with domain u 1, v 2π, as a double integral in u and v. o not evaluate the integral. In
More informationProblem Set 5 Math 213, Fall 2016
Problem Set 5 Math 213, Fall 216 Directions: Name: Show all your work. You are welcome and encouraged to use Mathematica, or similar software, to check your answers and aid in your understanding of the
More information(a) The points (3, 1, 2) and ( 1, 3, 4) are the endpoints of a diameter of a sphere.
MATH 4 FINAL EXAM REVIEW QUESTIONS Problem. a) The points,, ) and,, 4) are the endpoints of a diameter of a sphere. i) Determine the center and radius of the sphere. ii) Find an equation for the sphere.
More information( ) ( ) ( ) ( ) Calculus III - Problem Drill 24: Stokes and Divergence Theorem
alculus III - Problem Drill 4: tokes and Divergence Theorem Question No. 1 of 1 Instructions: (1) Read the problem and answer choices carefully () Work the problems on paper as needed () Pick the 1. Use
More information1. (16 points) Write but do not evaluate the following integrals:
MATH xam # Solutions. (6 points) Write but do not evaluate the following integrals: (a) (6 points) A clindrical integral to calculate the volume of the solid which lies in the first octant (where x,, and
More informationln e 2s+2t σ(m) = 1 + h 2 x + h 2 yda = dA = 90 da R
olution to et 5, Friday ay 7th ection 5.6: 15, 17. ection 5.7:, 5, 7, 16. (1) (ection 5.5, Problem ) Find a parametrization of the suface + y 9 between z and z. olution: cost, y sint and z s with t π and
More informationContents. MATH 32B-2 (18W) (L) G. Liu / (TA) A. Zhou Calculus of Several Variables. 1 Multiple Integrals 3. 2 Vector Fields 9
MATH 32B-2 (8W) (L) G. Liu / (TA) A. Zhou Calculus of Several Variables Contents Multiple Integrals 3 2 Vector Fields 9 3 Line and Surface Integrals 5 4 The Classical Integral Theorems 9 MATH 32B-2 (8W)
More informationOne side of each sheet is blank and may be used as scratch paper.
Math 244 Spring 2017 (Practice) Final 5/11/2017 Time Limit: 2 hours Name: No calculators or notes are allowed. One side of each sheet is blank and may be used as scratch paper. heck your answers whenever
More informationMath Exam IV - Fall 2011
Math 233 - Exam IV - Fall 2011 December 15, 2011 - Renato Feres NAME: STUDENT ID NUMBER: General instructions: This exam has 16 questions, each worth the same amount. Check that no pages are missing and
More informationMAT 211 Final Exam. Fall Jennings.
MAT 211 Final Exam. Fall 218. Jennings. Useful formulas polar coordinates spherical coordinates: SHOW YOUR WORK! x = rcos(θ) y = rsin(θ) da = r dr dθ x = ρcos(θ)cos(φ) y = ρsin(θ)cos(φ) z = ρsin(φ) dv
More information14.1. Multiple Integration. Iterated Integrals and Area in the Plane. Iterated Integrals. Iterated Integrals. MAC2313 Calculus III - Chapter 14
14 Multiple Integration 14.1 Iterated Integrals and Area in the Plane Objectives Evaluate an iterated integral. Use an iterated integral to find the area of a plane region. Copyright Cengage Learning.
More informationStudent name: Student ID: Math 265 (Butler) Midterm III, 10 November 2011
Student name: Student ID: Math 265 (Butler) Midterm III, November 2 This test is closed book and closed notes. No calculator is allowed for this test. For full credit show all of your work (legibly!).
More informationMATH 0350 PRACTICE FINAL FALL 2017 SAMUEL S. WATSON. a c. b c.
MATH 35 PRACTICE FINAL FALL 17 SAMUEL S. WATSON Problem 1 Verify that if a and b are nonzero vectors, the vector c = a b + b a bisects the angle between a and b. The cosine of the angle between a and c
More informationMcGill University April 20, Advanced Calculus for Engineers
McGill University April 0, 016 Faculty of Science Final examination Advanced Calculus for Engineers Math 64 April 0, 016 Time: PM-5PM Examiner: Prof. R. Choksi Associate Examiner: Prof. A. Hundemer Student
More informationM273Q Multivariable Calculus Spring 2017 Review Problems for Exam 3
M7Q Multivariable alculus Spring 7 Review Problems for Exam Exam covers material from Sections 5.-5.4 and 6.-6. and 7.. As you prepare, note well that the Fall 6 Exam posted online did not cover exactly
More informationMa 1c Practical - Solutions to Homework Set 7
Ma 1c Practical - olutions to omework et 7 All exercises are from the Vector Calculus text, Marsden and Tromba (Fifth Edition) Exercise 7.4.. Find the area of the portion of the unit sphere that is cut
More informationMath Review for Exam 3
1. ompute oln: (8x + 36xy)ds = Math 235 - Review for Exam 3 (8x + 36xy)ds, where c(t) = (t, t 2, t 3 ) on the interval t 1. 1 (8t + 36t 3 ) 1 + 4t 2 + 9t 4 dt = 2 3 (1 + 4t2 + 9t 4 ) 3 2 1 = 2 3 ((14)
More informationPractice Final Solutions
Practice Final Solutions Math 1, Fall 17 Problem 1. Find a parameterization for the given curve, including bounds on the parameter t. Part a) The ellipse in R whose major axis has endpoints, ) and 6, )
More informationSolutions to Sample Questions for Final Exam
olutions to ample Questions for Final Exam Find the points on the surface xy z 3 that are closest to the origin. We use the method of Lagrange Multipliers, with f(x, y, z) x + y + z for the square of the
More information1 + f 2 x + f 2 y dy dx, where f(x, y) = 2 + 3x + 4y, is
1. The value of the double integral (a) 15 26 (b) 15 8 (c) 75 (d) 105 26 5 4 0 1 1 + f 2 x + f 2 y dy dx, where f(x, y) = 2 + 3x + 4y, is 2. What is the value of the double integral interchange the order
More information1. If the line l has symmetric equations. = y 3 = z+2 find a vector equation for the line l that contains the point (2, 1, 3) and is parallel to l.
. If the line l has symmetric equations MA 6 PRACTICE PROBLEMS x = y = z+ 7, find a vector equation for the line l that contains the point (,, ) and is parallel to l. r = ( + t) i t j + ( + 7t) k B. r
More informationMath 53 Spring 2018 Practice Midterm 2
Math 53 Spring 218 Practice Midterm 2 Nikhil Srivastava 8 minutes, closed book, closed notes 1. alculate 1 y 2 (x 2 + y 2 ) 218 dxdy Solution. Since the type 2 region D = { y 1, x 1 y 2 } is a quarter
More informationMath 11 Fall 2007 Practice Problem Solutions
Math 11 Fall 27 Practice Problem olutions Here are some problems on the material we covered since the second midterm. This collection of problems is not intended to mimic the final in length, content,
More informationPractice Problems for the Final Exam
Math 114 Spring 2017 Practice Problems for the Final Exam 1. The planes 3x + 2y + z = 6 and x + y = 2 intersect in a line l. Find the distance from the origin to l. (Answer: 24 3 ) 2. Find the area of
More informationMath 261 Solutions To Sample Exam 2 Problems
Solutions to Sample Eam Problems Math 6 Math 6 Solutions To Sample Eam Problems. Given to the right is the graph of a portion of four curves:,, and + 4. Note that these curves divide the plane into separate
More informationSection 14.1 Vector Functions and Space Curves
Section 14.1 Vector Functions and Space Curves Functions whose range does not consists of numbers A bulk of elementary mathematics involves the study of functions - rules that assign to a given input a
More information5. Triple Integrals. 5A. Triple integrals in rectangular and cylindrical coordinates. 2 + y + z x=0. y Outer: 1
5. Triple Integrals 5A. Triple integrals in rectangular and clindrical coordinates ] 5A- a) (x + + )dxdd Inner: x + x( + ) + + x ] ] Middle: + + + ( ) + Outer: + 6 x ] x b) x ddxd Inner: x x 3 4 ] ] +
More informationName: Date: 12/06/2018. M20550 Calculus III Tutorial Worksheet 11
1. ompute the surface integral M255 alculus III Tutorial Worksheet 11 x + y + z) d, where is a surface given by ru, v) u + v, u v, 1 + 2u + v and u 2, v 1. olution: First, we know x + y + z) d [ ] u +
More informationFigure 25:Differentials of surface.
2.5. Change of variables and Jacobians In the previous example we saw that, once we have identified the type of coordinates which is best to use for solving a particular problem, the next step is to do
More informationPage Problem Score Max Score a 8 12b a b 10 14c 6 6
Fall 14 MTH 34 FINAL EXAM December 8, 14 Name: PID: Section: Instructor: DO NOT WRITE BELOW THIS LINE. Go to the next page. Page Problem Score Max Score 1 5 5 1 3 5 4 5 5 5 6 5 7 5 8 5 9 5 1 5 11 1 3 1a
More informationProblem Set 6 Math 213, Fall 2016
Problem Set 6 Math 213, Fall 216 Directions: Name: Show all your work. You are welcome and encouraged to use Mathematica, or similar software, to check your answers and aid in your understanding of the
More informationAnswers and Solutions to Section 13.7 Homework Problems 1 19 (odd) S. F. Ellermeyer April 23, 2004
Answers and olutions to ection 1.7 Homework Problems 1 19 (odd). F. Ellermeyer April 2, 24 1. The hemisphere and the paraboloid both have the same boundary curve, the circle x 2 y 2 4. Therefore, by tokes
More informationPeter Alfeld Math , Fall 2005
WeBWorK assignment due 9/2/05 at :59 PM..( pt) Consider the parametric equation x = 2(cosθ + θsinθ) y = 2(sinθ θcosθ) What is the length of the curve for θ = 0 to θ = 7 6 π? 2.( pt) Let a = (-2 4 2) and
More informationMATH 228: Calculus III (FALL 2016) Sample Problems for FINAL EXAM SOLUTIONS
MATH 228: Calculus III (FALL 216) Sample Problems for FINAL EXAM SOLUTIONS MATH 228 Page 2 Problem 1. (2pts) Evaluate the line integral C xy dx + (x + y) dy along the parabola y x2 from ( 1, 1) to (2,
More informationHOMEWORK 8 SOLUTIONS
HOMEWOK 8 OLUTION. Let and φ = xdy dz + ydz dx + zdx dy. let be the disk at height given by: : x + y, z =, let X be the region in 3 bounded by the cone and the disk. We orient X via dx dy dz, then by definition
More informationProblem Points S C O R E
MATH 34F Final Exam March 19, 13 Name Student I # Your exam should consist of this cover sheet, followed by 7 problems. Check that you have a complete exam. Unless otherwise indicated, show all your work
More informationReview problems for the final exam Calculus III Fall 2003
Review problems for the final exam alculus III Fall 2003 1. Perform the operations indicated with F (t) = 2t ı 5 j + t 2 k, G(t) = (1 t) ı + 1 t k, H(t) = sin(t) ı + e t j a) F (t) G(t) b) F (t) [ H(t)
More information7a3 2. (c) πa 3 (d) πa 3 (e) πa3
1.(6pts) Find the integral x, y, z d S where H is the part of the upper hemisphere of H x 2 + y 2 + z 2 = a 2 above the plane z = a and the normal points up. ( 2 π ) Useful Facts: cos = 1 and ds = ±a sin
More informationEXAM 2 ANSWERS AND SOLUTIONS, MATH 233 WEDNESDAY, OCTOBER 18, 2000
EXAM 2 ANSWERS AND SOLUTIONS, MATH 233 WEDNESDAY, OCTOBER 18, 2000 This examination has 30 multiple choice questions. Problems are worth one point apiece, for a total of 30 points for the whole examination.
More informationMath 212-Lecture Integration in cylindrical and spherical coordinates
Math 22-Lecture 6 4.7 Integration in cylindrical and spherical coordinates Cylindrical he Jacobian is J = (x, y, z) (r, θ, z) = cos θ r sin θ sin θ r cos θ = r. Hence, d rdrdθdz. If we draw a picture,
More informationAnswer sheet: Final exam for Math 2339, Dec 10, 2010
Answer sheet: Final exam for Math 9, ec, Problem. Let the surface be z f(x,y) ln(y + cos(πxy) + e ). (a) Find the gradient vector of f f(x,y) y + cos(πxy) + e πy sin(πxy), y πx sin(πxy) (b) Evaluate f(,
More informationon an open connected region D, then F is conservative on D. (c) If curl F=curl G on R 3, then C F dr = C G dr for all closed path C.
. (5%) Determine the statement is true ( ) or false ( ). 微甲 -4 班期末考解答和評分標準 (a) If f(x, y) is continuous on the rectangle R = {(x, y) a x b, c y d} except for finitely many points, then f(x, y) is integrable
More informationMath 23b Practice Final Summer 2011
Math 2b Practice Final Summer 211 1. (1 points) Sketch or describe the region of integration for 1 x y and interchange the order to dy dx dz. f(x, y, z) dz dy dx Solution. 1 1 x z z f(x, y, z) dy dx dz
More informationMath 20C Homework 2 Partial Solutions
Math 2C Homework 2 Partial Solutions Problem 1 (12.4.14). Calculate (j k) (j + k). Solution. The basic properties of the cross product are found in Theorem 2 of Section 12.4. From these properties, we
More informationSolutions to the Final Exam, Math 53, Summer 2012
olutions to the Final Exam, Math 5, ummer. (a) ( points) Let be the boundary of the region enclosedby the parabola y = x and the line y = with counterclockwise orientation. alculate (y + e x )dx + xdy.
More informationMath 31CH - Spring Final Exam
Math 3H - Spring 24 - Final Exam Problem. The parabolic cylinder y = x 2 (aligned along the z-axis) is cut by the planes y =, z = and z = y. Find the volume of the solid thus obtained. Solution:We calculate
More informationUse partial integration with respect to y to compute the inner integral (treating x as a constant.)
Math 54 ~ Multiple Integration 4. Iterated Integrals and Area in the Plane Iterated Integrals f ( x, y) dydx = f ( x, y) dy dx b g ( x) b g ( x) a g ( x) a g ( x) Use partial integration with respect to
More informationCreated by T. Madas SURFACE INTEGRALS. Created by T. Madas
SURFACE INTEGRALS Question 1 Find the area of the plane with equation x + 3y + 6z = 60, 0 x 4, 0 y 6. 8 Question A surface has Cartesian equation y z x + + = 1. 4 5 Determine the area of the surface which
More informationMath 11 Fall 2016 Final Practice Problem Solutions
Math 11 Fall 216 Final Practice Problem olutions Here are some problems on the material we covered since the second midterm. This collection of problems is not intended to mimic the final in length, content,
More informationJUST THE MATHS UNIT NUMBER INTEGRATION APPLICATIONS 10 (Second moments of an arc) A.J.Hobson
JUST THE MATHS UNIT NUMBER 13.1 INTEGRATION APPLICATIONS 1 (Second moments of an arc) by A.J.Hobson 13.1.1 Introduction 13.1. The second moment of an arc about the y-axis 13.1.3 The second moment of an
More informationMA 351 Fall 2007 Exam #1 Review Solutions 1
MA 35 Fall 27 Exam # Review Solutions THERE MAY BE TYPOS in these solutions. Please let me know if you find any.. Consider the two surfaces ρ 3 csc θ in spherical coordinates and r 3 in cylindrical coordinates.
More informationLet s estimate the volume under this surface over the rectangle R = [0, 4] [0, 2] in the xy-plane.
Math 54 - Vector Calculus Notes 3. - 3. Double Integrals Consider f(x, y) = 8 x y. Let s estimate the volume under this surface over the rectangle R = [, 4] [, ] in the xy-plane. Here is a particular estimate:
More informationPractice Problems: Exam 2 MATH 230, Spring 2011 Instructor: Dr. Zachary Kilpatrick Show all your work. Simplify as much as possible.
Practice Problems: Exam MATH, Spring Instructor: Dr. Zachary Kilpatrick Show all your work. Simplify as much as possible.. Write down a table of x and y values associated with a few t values. Then, graph
More informationSept , 17, 23, 29, 37, 41, 45, 47, , 5, 13, 17, 19, 29, 33. Exam Sept 26. Covers Sept 30-Oct 4.
MATH 23, FALL 2013 Text: Calculus, Early Transcendentals or Multivariable Calculus, 7th edition, Stewart, Brooks/Cole. We will cover chapters 12 through 16, so the multivariable volume will be fine. WebAssign
More informationMath 53 Final Exam, Prof. Srivastava May 11, 2018, 11:40pm 2:30pm, 155 Dwinelle Hall.
Math 53 Final Exam, Prof. Srivastava May 11, 2018, 11:40pm 2:30pm, 155 Dwinelle Hall. Name: SID: GSI: Name of the student to your left: Name of the student to your right: Instructions: Write all answers
More informationMATHEMATICS 200 April 2010 Final Exam Solutions
MATHEMATICS April Final Eam Solutions. (a) A surface z(, y) is defined by zy y + ln(yz). (i) Compute z, z y (ii) Evaluate z and z y in terms of, y, z. at (, y, z) (,, /). (b) A surface z f(, y) has derivatives
More informationSolutions to old Exam 3 problems
Solutions to old Exam 3 problems Hi students! I am putting this version of my review for the Final exam review here on the web site, place and time to be announced. Enjoy!! Best, Bill Meeks PS. There are
More information14.7 Triple Integrals In Cylindrical and Spherical Coordinates Contemporary Calculus TRIPLE INTEGRALS IN CYLINDRICAL AND SPHERICAL COORDINATES
14.7 Triple Integrals In Cylindrical and Spherical Coordinates Contemporary Calculus 1 14.7 TIPLE INTEGALS IN CYLINDICAL AND SPHEICAL COODINATES In physics everything is straight, flat or round. Statement
More informationReview for the Final Test
Math 7 Review for the Final Test () Decide if the limit exists and if it exists, evaluate it. lim (x,y,z) (0,0,0) xz. x +y +z () Use implicit differentiation to find z if x + y z = 9 () Find the unit tangent
More informationMath 32B Discussion Session Week 10 Notes March 14 and March 16, 2017
Math 3B iscussion ession Week 1 Notes March 14 and March 16, 17 We ll use this week to review for the final exam. For the most part this will be driven by your questions, and I ve included a practice final
More informationMath 323 Exam 1 Practice Problem Solutions
Math Exam Practice Problem Solutions. For each of the following curves, first find an equation in x and y whose graph contains the points on the curve. Then sketch the graph of C, indicating its orientation.
More informationMath 2433 Notes Week Triple Integrals. Integration over an arbitrary solid: Applications: 1. Volume of hypersolid = f ( x, y, z ) dxdydz
Math 2433 Notes Week 11 15.6 Triple Integrals Integration over an arbitrary solid: Applications: 1. Volume of hypersolid = f ( x, y, z ) dxdydz S 2. Volume of S = dxdydz S Reduction to a repeated integral
More informationReview Sheet for the Final
Review Sheet for the Final Math 6-4 4 These problems are provided to help you study. The presence of a problem on this handout does not imply that there will be a similar problem on the test. And the absence
More informationMath 210, Final Exam, Practice Fall 2009 Problem 1 Solution AB AC AB. cosθ = AB BC AB (0)(1)+( 4)( 2)+(3)(2)
Math 2, Final Exam, Practice Fall 29 Problem Solution. A triangle has vertices at the points A (,,), B (, 3,4), and C (2,,3) (a) Find the cosine of the angle between the vectors AB and AC. (b) Find an
More informationPrint Your Name: Your Section:
Print Your Name: Your Section: Mathematics 1c. Practice Final Solutions This exam has ten questions. J. Marsden You may take four hours; there is no credit for overtime work No aids (including notes, books,
More informationPractice problems **********************************************************
Practice problems I will not test spherical and cylindrical coordinates explicitly but these two coordinates can be used in the problems when you evaluate triple integrals. 1. Set up the integral without
More informationWORKSHEET #13 MATH 1260 FALL 2014
WORKSHEET #3 MATH 26 FALL 24 NOT DUE. Short answer: (a) Find the equation of the tangent plane to z = x 2 + y 2 at the point,, 2. z x (, ) = 2x = 2, z y (, ) = 2y = 2. So then the tangent plane equation
More informationSections minutes. 5 to 10 problems, similar to homework problems. No calculators, no notes, no books, no phones. No green book needed.
MTH 34 Review for Exam 4 ections 16.1-16.8. 5 minutes. 5 to 1 problems, similar to homework problems. No calculators, no notes, no books, no phones. No green book needed. Review for Exam 4 (16.1) Line
More informationThe Volume of a Hypersphere
The hypersphere has the equation The Volume of a Hypersphere x 2 y 2 x 2 w 2 = 2 if centered at the origin (,,,) and has a radius of in four dimensional space. We approach the project of determining its
More informationTriple Integrals. y x
Triple Integrals. (a) If is an solid (in space), what does the triple integral dv represent? Wh? (b) Suppose the shape of a solid object is described b the solid, and f(,, ) gives the densit of the object
More informationMa 227 Final Exam Solutions 12/13/11
Ma 7 Final Exam Solutions /3/ Name: Lecture Section: (A and B: Prof. Levine, C: Prof. Brady) Problem a) ( points) Find the eigenvalues and eigenvectors of the matrix A. A 3 5 Solution. First we find the
More informationMATH 255 Applied Honors Calculus III Winter Homework 11. Due: Monday, April 18, 2011
MATH 255 Applied Honors Calculus III Winter 211 Homework 11 ue: Monday, April 18, 211 ection 17.7, pg. 1155: 5, 13, 19, 24. ection 17.8, pg. 1161: 3, 7, 13, 17 ection 17.9, pg. 1168: 3, 7, 19, 25. 17.7
More informationThe Divergence Theorem
Math 1a The Divergence Theorem 1. Parameterize the boundary of each of the following with positive orientation. (a) The solid x + 4y + 9z 36. (b) The solid x + y z 9. (c) The solid consisting of all points
More information(You may need to make a sin / cos-type trigonometric substitution.) Solution.
MTHE 7 Problem Set Solutions. As a reminder, a torus with radii a and b is the surface of revolution of the circle (x b) + z = a in the xz-plane about the z-axis (a and b are positive real numbers, with
More informationMULTIVARIABLE INTEGRATION
MULTIVARIABLE INTEGRATION (SPHERICAL POLAR COORDINATES) Question 1 a) Determine with the aid of a diagram an expression for the volume element in r, θ, ϕ. spherical polar coordinates, ( ) [You may not
More informationFinal exam (practice 1) UCLA: Math 32B, Spring 2018
Instructor: Noah White Date: Final exam (practice 1) UCLA: Math 32B, Spring 218 This exam has 7 questions, for a total of 8 points. Please print your working and answers neatly. Write your solutions in
More informationSolutions for the Practice Final - Math 23B, 2016
olutions for the Practice Final - Math B, 6 a. True. The area of a surface is given by the expression d, and since we have a parametrization φ x, y x, y, f x, y with φ, this expands as d T x T y da xy
More informationMATH 280 Multivariate Calculus Fall Integration over a surface. da. A =
MATH 28 Multivariate Calculus Fall 212 Integration over a surface Given a surface S in space, we can (conceptually) break it into small pieces each of which has area da. In me cases, we will add up these
More informationLine and Surface Integrals. Stokes and Divergence Theorems
Math Methods 1 Lia Vas Line and urface Integrals. tokes and Divergence Theorems Review of urves. Intuitively, we think of a curve as a path traced by a moving particle in space. Thus, a curve is a function
More informationMath 221 Examination 2 Several Variable Calculus
Math Examination Spring Instructions These problems should be viewed as essa questions. Before making a calculation, ou should explain in words what our strateg is. Please write our solutions on our own
More informationFinal Review Worksheet
Score: Name: Final Review Worksheet Math 2110Q Fall 2014 Professor Hohn Answers (in no particular order): f(x, y) = e y + xe xy + C; 2; 3; e y cos z, e z cos x, e x cos y, e x sin y e y sin z e z sin x;
More information