Math 212-Lecture Integration in cylindrical and spherical coordinates
|
|
- Winfred Allison
- 5 years ago
- Views:
Transcription
1 Math 22-Lecture 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, we can see directly that dv is really rdrdθdz. Spherical he Jacobian is (x, y, z) J = (ρ, φ, θ) = sin φ cos θ ρ cos φ cos θ ρ sin φ sin θ sin φ sin θ ρ cos φ sin θ ρ sin φ cos θ cos φ ρ sin φ = ρ2 sin φ. Hence, d ρ 2 sin φdρdφdθ. If we draw a picture, we can see clearly that this is true. Example: Find the centroid of the first octant portion of the ball x 2 +y 2 +z 2 a 2 using both cylindrical coordinates and spherical coordinates, assuming the density is uniform. Solution. In Cylindrical way: the sphere is r 2 + z 2 = a 2. Hence, we can have r a, θ π/2, z a 2 r 2. Due to the symmetry, we must have x = ȳ = z. hen, z = zδd δdv dv zdv = a π/2 a 2 r 2 rdzdθdr a π/2 a 2 r 2 zrdzdθdr
2 In Spherical way: the sphere is ρ = a. Hence, ρ a, φ π/2, θ π/2. zd ρ 2 sin φdθdφdρ. ρ cos φ ρ 2 sin φdθdφdρ = Example: Write out the region bounded by z = x 2 + y 2 and z = y in cylindrical coordinates. Solution. We have done this example before. In cylindrical, they are z = r 2 and z = r sin θ. he intersection is r = sin θ. he projection onto xy plane is a circle. For θ, we set r =, and see, π are two adjacent zeros. Hence, θ π, r sin θ, r 2 z r sin θ. Example: Set up the integral for the area inside the two circles r = and r = 2 sin θ. Set up the integral for the volume of the solid bounded by r =, r = 2 sin θ, z = y and the xy plane. Solution. For the area, A = R da. In polar, da = rdrdθ. We see that we must divide the integral into three pieces. = 2 sin θ. We find θ = π/6 and θ = 5π/6. Hence, A = π/6 2 sin θ rdrdθ + 5π/6 π/6 he volume is R (z 2 z )da = R yda. Hence, π/6 2 sin θ r sin θrdrdθ+ 5π/6 π/6 π 2 sin θ rdrdθ + rdrdθ. 5π/6 ρ 3 sin φ cos φdθdφdρ. π 2 sin θ r sin θrdrdθ+ r sin θrdrdθ. 5π/6 Example: Set up the integral for the volume bounded by x 2 +y 2 +z 2 = 4 and x 2 + y 2 2x =. 2
3 Solution. We use cylindrical coordinates. r 2 + z 2 = 4 and r 2 cos θ =. he region is determined by r 2 cos θ =. We set r = and have cos θ =. Hence, π/2 θ π/2. hen, r 2 cos θ. For z, we find 4 r 2 z 4 r 2. Note d rdrdθdz. he volume is therefore: d π/2 2 cos θ 4 r 2 π/2 4 r 2 rdzdrdθ Example: Set up the integral for the mass of the region contained in the sphere x 2 + y 2 + (z a) 2 = a 2 but below z = r with unit density. If we draw the picture, we see that the spherical coordinates are the best. Solution. z = r is just φ = π/4. he sphere is ρ 2 2aρ cos φ = or ρ = 2a cos φ. Hence, we have π/4 φ π/2, ρ 2a cos φ, θ < 2π. hen, m = π/2 2a cos φ 2π π/4 ρ 2 sin φdθdρdφ. Example: Consider the ice-cream cone above φ = π/6 but below ρ = 2a cos φ. Suppose the density is δ =. Set up the integrals for the total mass and centroid. Solution. his problem is convenient in spherical coordinates. φ π/6, θ < 2π, ρ 2a cos φ. he volume element is d ρ 2 sin φdρdθdφ. he total mass is π/6 2π 2a cos φ m = δd ρ 2 sin φdρdθdφ. For the centroid, we use symmetry do conclude that x = ȳ =. hen, z = m because z = ρ cos φ. δzd m π/6 2π 2a cos φ ρ cos φρ 2 sin φdρdθdφ 3
4 4.8 Surface Area Previously, we see that a vector valued function with a single variable r(t) is a curve in space. Now, if the function is vector-valued but has two variables (parameters) r(u, v) = x(u, v), y(u, v), z(u, v), where r again is the position vector of the point, what will the object be? Fixing v = v, r(u, v ) is a curve. Now, for different v = v, it s another curve. he object is thus a family of curves, and they form a surface. Example: he graph z = f(x, y) is a surface in 3D space. Parametrize it. r = x, y, z. We choose x, y as the parameters. hen, r = x, y, f(x, y). Example: Let ρ, φ, θ be the spherical coordinates. he function ρ = h(φ, θ) gives a surface in the space. (Example is ρ = 2.) Parametrize this surface. Solution. r = x, y, z = h(φ, θ) sin φ cos θ, h(φ, θ) sin φ sin θ, h(φ, θ) cos φ. Example: Parametrize the rectangle x 2, y 3, z =. Solution. hence his is a special case of the first example, z = f(x, y) = and r(x, y) = x, y,, x 2, y 3 Given a parametric surface r(u, v) = x(u, v)i + y(u, v)j + z(u, v)k = x(u, v), y(u, v), z(u, v), we call it smooth if r u = x u, y u, z u, r v = x v, y v, z v, are both nonzero and nonparallel. Consider the small area for the rectangle u v in u-v plane. It s a parallelogram on the surface under the mapping r(u, v). Draw a picture. One edge is a = r(u + u, v) r(u, v) r u u. he other edge is similarly b r v v. he area is therefore S a b r u r v u v. 4
5 N = r u r v is a normal vector of the surface. he total area is A = a(s) = ds = r u r v dudv. u,v u,v ds = r u r v dudv is the surface area element and ds = N N ds = Ndudv = r u r v dudv is the directed surface area element. Here, we see that r u r v plays the same role as the Jacobian in the change of variables for double integrals. It s the amplification factor between the areas. Example: If u, v are the Cartesian coordinates x, y, then r = x, y, f(x, y). It is the graph of z = f(x, y). r x r y = f x. f y,. his makes sense as it is just F where F = z f(x, y). he area is A = R + f 2 x + f 2 y dxdy. We compute the area of the ellipse cut from z = 2x + 2y + by x 2 + y 2 =. What if the surface is the one cut from x = 2y + 2z + from y + z =, y =, z =? Example: Find the area of the spiral ramp z = θ, r, θ π. Solution. We parametrize the surface r(r, θ) = r cos θ, r sin θ, θ. r r r θ = cos θ, sin θ, r sin θ, r cos θ, he magnitude of this is + r 2 he integral is π π/4 + r 2 dθdr = π sec 3 θdθ = π 2 ( 2 + ln( + 2)) Exercise. Compute the area of the portion of z 2 = 3(x 2 + y 2 ) below z = 3, and above xy plane. r = r cos θ, r sin θ, 3r. r 3, θ < 2π. he magnitude of r r r θ is 2r. hen, 3 2π 2rdθdr 5
Practice problems. m zδdv. In our case, we can cancel δ and have z =
Practice problems 1. Consider a right circular cone of uniform density. The height is H. Let s say the distance of the centroid to the base is d. What is the value d/h? We can create a coordinate system
More informationDouble Integrals. Advanced Calculus. Lecture 2 Dr. Lahcen Laayouni. Department of Mathematics and Statistics McGill University.
Lecture Department of Mathematics and Statistics McGill University January 9, 7 Polar coordinates Change of variables formula Polar coordinates In polar coordinates, we have x = r cosθ, r = x + y y = r
More informationPractice problems. 1. Evaluate the double or iterated integrals: First: change the order of integration; Second: polar.
Practice problems 1. Evaluate the double or iterated integrals: x 3 + 1dA where = {(x, y) : 0 y 1, y x 1}. 1/ 1 y 0 3y sin(x + y )dxdy First: change the order of integration; Second: polar.. Consider the
More information********************************************************** 1. Evaluate the double or iterated integrals:
Practice problems 1. (a). Let f = 3x 2 + 4y 2 + z 2 and g = 2x + 3y + z = 1. Use Lagrange multiplier to find the extrema of f on g = 1. Is this a max or a min? No max, but there is min. Hence, among the
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 informationMath Vector Calculus II
Math 255 - Vector Calculus II Review Notes Vectors We assume the reader is familiar with all the basic concepts regarding vectors and vector arithmetic, such as addition/subtraction of vectors in R n,
More informationTopic 5.6: Surfaces and Surface Elements
Math 275 Notes Topic 5.6: Surfaces and Surface Elements Textbook Section: 16.6 From the Toolbox (what you need from previous classes): Using vector valued functions to parametrize curves. Derivatives of
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 informationPractice problems ********************************************************** 1. Divergence, curl
Practice problems 1. Set up the integral without evaluation. The volume inside (x 1) 2 + y 2 + z 2 = 1, below z = 3r but above z = r. This problem is very tricky in cylindrical or Cartesian since we must
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 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. 1. Evaluate the double or iterated integrals: First: change the order of integration; Second: polar.
Practice problems 1. Evaluate the double or iterated integrals: R x 3 + 1dA where R = {(x, y) : 0 y 1, y x 1}. 1/ 1 y 0 3y sin(x + y )dxdy First: change the order of integration; Second: polar.. Consider
More informationArchive 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 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 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 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 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 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 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 informationSummary for Vector Calculus and Complex Calculus (Math 321) By Lei Li
Summary for Vector alculus and omplex alculus (Math 321) By Lei Li 1 Vector alculus 1.1 Parametrization urves, surfaces, or volumes can be parametrized. Below, I ll talk about 3D case. Suppose we use e
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 informationMATHS 267 Answers to Stokes Practice Dr. Jones
MATH 267 Answers to tokes Practice Dr. Jones 1. Calculate the flux F d where is the hemisphere x2 + y 2 + z 2 1, z > and F (xz + e y2, yz, z 2 + 1). Note: the surface is open (doesn t include any of the
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 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 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 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 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 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 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 informationCalculus III 2004 Summer Practice Final 8/3/2004
.. Calculus III 4 ummer Practice Final 8/3/4. Compute the following limits if they exist: (a) lim (x,y) (,) e xy x+. cos x (b) lim x. (x,y) (,) x 4 +y 4 (a) ince lim (x,y) (,) exy and lim x + 6 in a (x,y)
More informationMathematics 205 Solutions for HWK 23. e x2 +y 2 dxdy
Mathematics 5 Solutions for HWK Problem 1. 6. p7. Let D be the unit disk: x + y 1. Evaluate the integral e x +y dxdy by making a change of variables to polar coordinates. D Solution. Step 1. The integrand,
More informationLaplace equation in polar coordinates
Laplace equation in polar coordinates The Laplace equation is given by 2 F 2 + 2 F 2 = 0 We have x = r cos θ, y = r sin θ, and also r 2 = x 2 + y 2, tan θ = y/x We have for the partials with respect to
More informationName: Instructor: Lecture time: TA: Section time:
Math 222 Final May 11, 29 Name: Instructor: Lecture time: TA: Section time: INSTRUCTIONS READ THIS NOW This test has 1 problems on 16 pages worth a total of 2 points. Look over your test package right
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 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 informationf(p i )Area(T i ) F ( r(u, w) ) (r u r w ) da
MAH 55 Flux integrals Fall 16 1. Review 1.1. Surface integrals. Let be a surface in R. Let f : R be a function defined on. efine f ds = f(p i Area( i lim mesh(p as a limit of Riemann sums over sampled-partitions.
More informationCOMPLETE Chapter 15 Multiple Integrals. Section 15.1 Double Integrals Over Rectangles. Section 15.2 Iterated Integrals
Mat 7 Calculus III Updated on /3/7 Dr. Firoz COMPLT Chapter 5 Multiple Integrals Section 5. Double Integrals Over ectangles amples:. valuate the iterated integral a) (5 ) da, {(, ), } and b) (4 ) da, [,]
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 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 informationSOLUTIONS TO THE FINAL EXAM. December 14, 2010, 9:00am-12:00 (3 hours)
SOLUTIONS TO THE 18.02 FINAL EXAM BJORN POONEN December 14, 2010, 9:00am-12:00 (3 hours) 1) For each of (a)-(e) below: If the statement is true, write TRUE. If the statement is false, write FALSE. (Please
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 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 information51. General Surface Integrals
51. General urface Integrals The area of a surface in defined parametrically by r(u, v) = x(u, v), y(u, v), z(u, v) over a region of integration in the input-variable plane is given by d = r u r v da.
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 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 234 Exam 3 Review Sheet
Math 234 Exam 3 Review Sheet Jim Brunner LIST OF TOPIS TO KNOW Vector Fields lairaut s Theorem & onservative Vector Fields url Divergence Area & Volume Integrals Using oordinate Transforms hanging the
More informationMATH 52 FINAL EXAM DECEMBER 7, 2009
MATH 52 FINAL EXAM DECEMBER 7, 2009 THIS IS A CLOSED BOOK, CLOSED NOTES EXAM. NO CALCULATORS OR OTHER ELECTRONIC DEVICES ARE PERMITTED. IF YOU NEED EXTRA SPACE, PLEASE USE THE BACK OF THE PREVIOUS PROB-
More informatione x2 dxdy, e x2 da, e x2 x 3 dx = e
STS26-4 Calculus II: The fourth exam Dec 15, 214 Please show all your work! Answers without supporting work will be not given credit. Write answers in spaces provided. You have 1 hour and 2minutes to complete
More informationDr. Allen Back. Nov. 5, 2014
Dr. Allen Back Nov. 5, 2014 12 lectures, 4 recitations left including today. a Most of what remains is vector integration and the integral theorems. b We ll start 7.1, 7.2,4.2 on Friday. c If you are not
More informationSOLUTIONS TO HOMEWORK ASSIGNMENT #2, Math 253
SOLUTIONS TO HOMEWORK ASSIGNMENT #, Math 5. Find the equation of a sphere if one of its diameters has end points (, 0, 5) and (5, 4, 7). The length of the diameter is (5 ) + ( 4 0) + (7 5) = =, so the
More information( ) ( ) Math 17 Exam II Solutions
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
More informationFigure 21:The polar and Cartesian coordinate systems.
Figure 21:The polar and Cartesian coordinate systems. Coordinate systems in R There are three standard coordinate systems which are used to describe points in -dimensional space. These coordinate systems
More informationPRACTICE PROBLEMS. Please let me know if you find any mistakes in the text so that i can fix them. 1. Mixed partial derivatives.
PRACTICE PROBLEMS Please let me know if you find any mistakes in the text so that i can fix them. 1.1. Let Show that f is C 1 and yet How is that possible? 1. Mixed partial derivatives f(x, y) = {xy x
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 informationName: SOLUTIONS Date: 11/9/2017. M20550 Calculus III Tutorial Worksheet 8
Name: SOLUTIONS Date: /9/7 M55 alculus III Tutorial Worksheet 8. ompute R da where R is the region bounded by x + xy + y 8 using the change of variables given by x u + v and y v. Solution: We know R is
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 informationRule ST1 (Symmetry). α β = β α for 1-forms α and β. Like the exterior product, the symmetric tensor product is also linear in each slot :
2. Metrics as Symmetric Tensors So far we have studied exterior products of 1-forms, which obey the rule called skew symmetry: α β = β α. There is another operation for forming something called the symmetric
More informationProblem 1. Use a line integral to find the plane area enclosed by the curve C: r = a cos 3 t i + b sin 3 t j (0 t 2π). Solution: We assume a > b > 0.
MATH 64: FINAL EXAM olutions Problem 1. Use a line integral to find the plane area enclosed by the curve C: r = a cos 3 t i + b sin 3 t j ( t π). olution: We assume a > b >. A = 1 π (xy yx )dt = 3ab π
More informationMath 32B Discussion Session Session 3 Notes August 14, 2018
Math 3B Discussion Session Session 3 Notes August 4, 8 In today s discussion we ll think about two common applications of multiple integrals: locating centers of mass and moments of inertia. Centers of
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 information4. Be able to set up and solve an integral using a change of variables. 5. Might be useful to remember the transformation formula for rotations.
Change of variables What to know. Be able to find the image of a transformation 2. Be able to invert a transformation 3. Be able to find the Jacobian of a transformation 4. Be able to set up and solve
More informationSolutions to Practice Exam 2
Solutions to Practice Eam Problem : For each of the following, set up (but do not evaluate) iterated integrals or quotients of iterated integral to give the indicated quantities: Problem a: The average
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 informationFinal exam (practice 1) UCLA: Math 32B, Spring 2018
Instructor: Noah White Date: Final exam (practice 1) UCLA: Math 32B, Spring 2018 This exam has 7 questions, for a total of 80 points. Please print your working and answers neatly. Write your solutions
More informationx + ye z2 + ze y2, y + xe z2 + ze x2, z and where T is the
1.(8pts) Find F ds where F = x + ye z + ze y, y + xe z + ze x, z and where T is the T surface in the pictures. (The two pictures are two views of the same surface.) The boundary of T is the unit circle
More informationMarking Scheme for the end semester examination of MTH101, (I) for n N. Show that (x n ) converges and find its limit. [5]
Marking Scheme for the end semester examination of MTH, 3-4 (I). (a) Let x =, x = and x n+ = xn+x for n N. Show that (x n ) converges and find its limit. [5] Observe that x n+ x = x x n [] The sequence
More informationCalculus III. Math 233 Spring Final exam May 3rd. Suggested solutions
alculus III Math 33 pring 7 Final exam May 3rd. uggested solutions This exam contains twenty problems numbered 1 through. All problems are multiple choice problems, and each counts 5% of your total score.
More information1. (a) (5 points) Find the unit tangent and unit normal vectors T and N to the curve. r (t) = 3 cos t, 0, 3 sin t, r ( 3π
1. a) 5 points) Find the unit tangent and unit normal vectors T and N to the curve at the point P 3, 3π, r t) 3 cos t, 4t, 3 sin t 3 ). b) 5 points) Find curvature of the curve at the point P. olution:
More information4.4 Change of Variable in Integrals: The Jacobian
4.4. CHANGE OF VAIABLE IN INTEGALS: THE JACOBIAN 4 4.4 Change of Variable in Integrals: The Jacobian In this section, we generalize to multiple integrals the substitution technique used with definite integrals.
More information234 Review Sheet 2 Solutions
4 Review Sheet Solutions. Find all the critical points of the following functions and apply the second derivative test. (a) f(x, y) (x y)(x + y) ( ) x f + y + (x y)x (x + y) + (x y) ( ) x + ( x)y x + x
More informationMATH H53 : Final exam
MATH H53 : Final exam 11 May, 18 Name: You have 18 minutes to answer the questions. Use of calculators or any electronic items is not permitted. Answer the questions in the space provided. If you run out
More informationis a surface above the xy-plane over R.
Chapter 13 Multiple Integration Section 13.1Double Integrals over ectangular egions ecall the Definite Integral from Chapter 5 b a n * lim i f x dx f x x n i 1 b If f x 0 then f xdx is the area under the
More information1 4 (1 cos(4θ))dθ = θ 4 sin(4θ)
M48M Final Exam Solutions, December 9, 5 ) A polar curve Let C be the portion of the cloverleaf curve r = sin(θ) that lies in the first quadrant a) Draw a rough sketch of C This looks like one quarter
More informationf dr. (6.1) f(x i, y i, z i ) r i. (6.2) N i=1
hapter 6 Integrals In this chapter we will look at integrals in more detail. We will look at integrals along a curve, and multi-dimensional integrals in 2 or more dimensions. In physics we use these integrals
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 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 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 332: Vector Analysis Summer 2005 Homework
MATH 332, (Vector Analysis), Summer 2005: Homework 1 Instructor: Ivan Avramidi MATH 332: Vector Analysis Summer 2005 Homework Set 1. (Scalar Product, Equation of a Plane, Vector Product) Sections: 1.9,
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 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 informationFinal Exam Review Sheet : Comments and Selected Solutions
MATH 55 Applied Honors alculus III Winter Final xam Review heet : omments and elected olutions Note: The final exam will cover % among topics in chain rule, linear approximation, maximum and minimum values,
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 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 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 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 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 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 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 information1.1 Single Variable Calculus versus Multivariable Calculus Rectangular Coordinate Systems... 4
MATH2202 Notebook 1 Fall 2015/2016 prepared by Professor Jenny Baglivo Contents 1 MATH2202 Notebook 1 3 1.1 Single Variable Calculus versus Multivariable Calculus................... 3 1.2 Rectangular Coordinate
More informationMTH101A (2016), Tentative Marking Scheme - End sem. exam
MTH11A (16), Tentative Marking Scheme - End sem. eam 1. (a) Let f(, y, z) = yz and S be + y + z = 6. Using Lagrange multipliers method, find the maimum and minimum values of f on S. [7] Lag. Eqns.: yz
More informationMTH 234 Exam 2 November 21st, Without fully opening the exam, check that you have pages 1 through 12.
Name: Section: Recitation Instructor: INSTRUCTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages 1 through 12. Show all your work on the standard
More informationMATH2111 Higher Several Variable Calculus Integration
MATH2 Higher Several Variable Calculus Integration Dr. Jonathan Kress School of Mathematics and Statistics University of New South Wales Semester, 26 [updated: April 3, 26] JM Kress (UNSW Maths & Stats)
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 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 informationLINE AND SURFACE INTEGRALS: A SUMMARY OF CALCULUS 3 UNIT 4
LINE AN URFAE INTEGRAL: A UMMARY OF ALULU 3 UNIT 4 The final unit of material in multivariable calculus introduces many unfamiliar and non-intuitive concepts in a short amount of time. This document attempts
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 informationQMUL, School of Physics and Astronomy Date: 18/01/2019
QMUL, School of Physics and stronomy Date: 8//9 PHY Mathematical Techniques Solutions for Exercise Class Script : Coordinate Systems and Double Integrals. Calculate the integral: where the region is defined
More informationMath 2374: Multivariable Calculus and Vector Analysis
Math 2374: Multivariable Calulus and Vetor Analysis Part 26 Fall 2012 The integrals of multivariable alulus line integral of salar-valued funtion line integral of vetor fields surfae integral of salar-valued
More informationWeek 7: Integration: Special Coordinates
Week 7: Integration: Special Coordinates Introduction Many problems naturally involve symmetry. One should exploit it where possible and this often means using coordinate systems other than Cartesian coordinates.
More information12.1. Cartesian Space
12.1. Cartesian Space In most of your previous math classes, we worked with functions on the xy-plane only meaning we were working only in 2D. Now we will be working in space, or rather 3D. Now we will
More information