Math 21a Homework 24 Solutions Spring, 2014
|
|
- Lindsey Holt
- 5 years ago
- Views:
Transcription
1 Math a Homework olutions pring, Due Friday, April th (MWF) or Tuesday, April 5th (TTh) This assignment is officially on urface Area (ection.6) and calar urface Integrals (ection.6), but it s most useful for practicing your parametrizations! You are encouraged to think about and draw the relevant surfaces and the grid lines of the parametrizations as practice.. Find the area of the given surface: (a) (tewart.6 #8 ) The helicoid (or spiral ramp) with vector equation r(u, v) = u cos v i + u sin v j + v k, and u and v π. (Note: the integral is doable, but very annoying. One strategy is: make an x = tan t substitution, and then integrate the result by parts with u = sec t. You must at least set up the integral correctly, but if you don t want to integrate it by hand, it s fine to use Mathematica s Integrate command (it works the same way as NIntegrate from HW ) to find an exact answer.) Recall that the surface area of this parmeterized surface is Area = π r u r v du dv. Here the partial derivatives are r u = cos v, sin v, and r v = u sin v, u cos v,, so their cross product is r u r v = sin v, cos v, u. This has length r u r v = u +. Thus the surface area is Area = π u + du dv = π u + dv du = π u + du. This is the integral described as very annoying in the problem statement. Mathematica gives us the answer + arcsinh() Area = π, and with a little investigation we can determine that arcsinh() = ln( + ). But how did we get there? Here we follow the hint in the problem statement and let u = tan(t), so du = sec (t) dt. The limits change from u = to to t = arctan() = to arctan() = π. We get Area = π π/ π/ tan t + sec t dt = π sec t dt (where we ve used the equality tan t + = sec t). Now this integral is a classic pain and we won t actually solve it except in that footnote. uffice it to say the answer is Area = π = π π/ sec t dt [ = π sec t tan t + [ ( ) = π + ( ) + ln( + ). This is the same as the answer from Mathematica. ] π/ ln sec t + tan t ( ln( + ) ln( + ) It has it s own Wikipedia page: Basically the idea is to integrate by parts with u = sec t and dv = sec t dt, so v = tan t. Then you get, after using the tan t + = sec t equality again, a formula like sec t dt = sec t tan t sec t dt + sec t dt. Then you use this the trick of adding sec t dt to both sides. This leaves us with the integral of sec t, which has it s own Wikipedia page: The general approach there is to know the answer already. This sounds like a joke, but really what you do is re-write sec t in a way that you only know to do because you ve seen the trick before: Combining these gives sec t dt = sec t + sec t tan t sec t + tan t dt = ln sec t + tan t + C. sec t dt = sec t tan t + ln sec t + tan t + C. ) ]
2 (b) (tewart.6 # ) The part of the paraboloid x = y + z that lies inside the cylinder y + z = 9. (This integral is quite doable by hand, and worth the practice.) In the non-standard cylindrical coordinates (x, r, θ) with x = x, y = r cos θ, z = r sin θ, the paraboloid can be parametrized by s(r, θ) = r, r cos θ, r sin θ θ < π, r. The partial derivatives are s r = r, cos θ, sin θ and s θ =, r sin θ, r cos θ, so their cross product is s r s θ = r, r cos θ, r sin θ and this has length s r s θ = 5 r. We are interested in the part of the paraboloid with r. The surface area of this part is s r s θ dr dθ = 5r dr dθ = π 5 r dr = 8π 5.. In this problem, we ll find the surface area of the part of the cylinder x +y = 9 that lies between the planes x+y+z = 6 and x + y + z = 5. (a) Use Mathematica to plot the cylinder and the planes: Plane = ContourPlotD[x+y+z == 5,{x,-,},{y,-,},{z,-,}] Plane = ContourPlotD[x+y+z == -6,{x,-,},{y,-,},{z,-,}] MyCylinder = ContourPlotD[x^+y^ == 9,{x,-,},{y,-,},{z,-,}] how[mycylinder, Plane,Plane] (b) Find a parametrization r (θ) of the curve C of intersection of the plane x+y +z = 5 with the cylinder x +y = 9. Also find a parametrization r (θ) of the curve C of intersection of the plane x + y + z = 6 with the cylinder. In both cases we parameterize x and y via x = cos θ, y = sin θ, then solve for z in terms of x and y, and thus in terms of θ. We get that C and C are parameterized by In both cases θ < π. r (θ) = cos θ, sin θ, 5 cos θ sin θ, r (θ) = cos θ, sin θ, 6 cos θ sin θ. (c) Write down a parametrization s(θ, u) of the part of the cylinder that lies between the two planes. The curves C and C should be two grid curves of the parametrization, and the bounds on the parameters should be of the form a u b and c θ d for constants a, b, c, d. One can take s(θ, u) = cos θ, sin θ, u cos θ sin θ with θ π and 6 u 5. (This is simply intersecting with the planes x + y + z = u from u = 6 to u = 5.)
3 (d) Use mathematica to plot your parametrization s(θ, u). (It s probably easiest to call the first parameter t instead of θ. You ll need to replace x(t,u), y(t,u) and z(t,u) by the components you found for s(t, u) in part (c), and the a,b,c,d by your bounds from part (c)): ParamCylinder = ParametricPlotD[{x(t,u),y(t,u),z(t,u)},{u,a,b},{t,c,d}] how[paramcylinder, Plane,Plane] (e) Use the parametrization you found to calculate the surface area of the part of the cylinder that lies between the two planes. The partial derivatives are s θ = sin θ, cos θ, sin θ cos θ and s u =,,. Hence s θ s u = cos θ, sin θ, =. The surface area is then equal to 5 s θ s u du dv = du dθ = π = 66π. (f) Does your answer from (e) make sense? Give an intuitive geometric reason that the surface area you found is the same as the surface area of a cylinder of radius and height. Every vertical segment that makes up the skewed cylinder has height. By sliding the segments so they all begin at a single height, we get a standard cylinder of height. This standard cylinder can be unwrapped to a cylinder of height and width 6π (the circumference of a circle of radius ), and thus this standard cylinder has surface area 66π.
4 . (Optional: 5 extra credit points) Evaluate the surface integral. (a) (tewart.6 #8 ) (x +y where is the surface with vector equation ) d r(u, v) = uv, u v, u + v, u + v (This integral is doable if you are careful about simplifying the integrand (and choosing a good coordinate system).) First let s simplify the integrand, as per the suggestion in the problem: x + y = (uv) + (u v ) = u v + u u v + v = u + u v + v = (u + v ). Then we compute r u r v in the usual way: r u = v, u, u, r v = u, v, v and so and Thus our surface integral is r u r u = 8uv, u v, u v r u r u = (8uv) + (u v ) + ( u v ) = 6u v + (6u u v + 6v ) + (6u + u v + 6v ) = u + 6u v + v = ( u + v ). (x + y ) d = u +v Changing to polar coordinates (u = r cos θ and v = r sin θ), we get (x + y ) d = (r ) (r ) r dr dθ = (u + v ) (u + v ) du dv. r 7 dr dθ = 8 π = π. (b) (tewart.6 # ) x +y +z d (Each of the integrals should be doable.) where is the part of the cylinder x + y = 9 between the planes z = and z =, together with the top and bottom disks. Note that has three parts to it: the sides of the cylinder which we ll call sides, the top top and the bottom bottom. We compute the integral separately for each piece. For sides, we use the cylindrical parametrization r(θ, z) = cos θ, sin θ, z with θ < π and z. ince the partial derivatives are r θ = sin θ, cos θ, and r z =,,, we see that the cross product is r θ r z = cos θ, sin θ, =. Therefore x + y + z d = sides (9 + z ) dz dθ = π [7z + z ] = π (5 + 8) = π. For bottom, we use the polar parameterization r(r, θ) = r cos(θ), r sin(θ), with r and θ < π. Then r r = cos(θ), sin(θ), and r θ = r sin(θ), r cos(θ),, so r r r θ =,, r and r r r θ = r. (This is just saying that in part of the xy-plane, the area element in polar coordinates is simply r dr dθ.) Thus bottom x + y + z d = r r dr dθ = r dr dθ = π = 8π.
5 Finally, for top, we use the polar parameterization r(r, θ) = r cos(θ), r sin(θ), with r and θ < π. Then r r and r θ are the same as with bottom, so we again get r r r θ = r. Hence top x + y + z d = (r + ) r dr dθ = ( ) (r + r) dr dθ = + π = 5π. Putting this all together, we get x + y + z d = x + y + z d + x + y + z d + x + y + z d sides bottom top = π + 8π + 5π = π.. (tewart.6 #8 ) Find the mass of a thin funnel in the shape of a cone z = x + y, z, if its density function is ρ(x, y, z) = z. We work in cylindrical coordinates, so our funnel is the graph of the function z(r, θ) = r. Thus we can parameterize this funnel as r(r, θ) = r cos(θ), r sin(θ), r ( θ < π and r ), so r r r θ = cos(θ), sin(θ), r sin(θ), r cos(θ), = r cos(θ), r sin(θ), r = r. Thus the funnel has mass equal to ( r) r dr dθ = π ] [5r r = π [( 8 6 ) ( 5 )] = 8π. 5. (Optional: extra credit points) (tewart.6 #6 ) The two cylinders y + z = and x + z = bound a solid region in R. The region is drawn on page 87. Find the surface area of the region. The region in question is made out of four identical pieces. One of them (the one that intersects the positive x-axis) can be parameterized by r(y, z) = z, y, z for y + z. Thus r y r z = z,,,, = z,, Therefore the surface area of this one piece is y +z dy dz = z z z = z z z dy dz. z. This is an improper integral (the denominator z = when z = ±), but we ll plow ahead and see what happens. The inner integral is straightforward, and we end up with something simple: dy dz = y y= z z z z dz = z dz = dz =. z y +z y= (Technically we should be taking limits as z approaches and to avoid the impropriety, but in the end we ll get for this integral.) ince this was the surface area of one of four identical pieces of the surface, the full surface has area () = 6. Reading: Please read ection. on vector fields, which we ll be discussing next class. If you are in the TTh section, please also skim through ection..
MATHS 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 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 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 informationCalculus II Practice Test 1 Problems: , 6.5, Page 1 of 10
Calculus II Practice Test Problems: 6.-6.3, 6.5, 7.-7.3 Page of This is in no way an inclusive set of problems there can be other types of problems on the actual test. To prepare for the test: review homework,
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 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 informationSection 7.4 #1, 5, 6, 8, 12, 13, 44, 53; Section 7.5 #7, 10, 11, 20, 22; Section 7.7 #1, 4, 10, 15, 22, 44
Math B Prof. Audrey Terras HW #4 Solutions Due Tuesday, Oct. 9 Section 7.4 #, 5, 6, 8,, 3, 44, 53; Section 7.5 #7,,,, ; Section 7.7 #, 4,, 5,, 44 7.4. Since 5 = 5 )5 + ), start with So, 5 = A 5 + B 5 +.
More informationMath 3435 Homework Set 11 Solutions 10 Points. x= 1,, is in the disk of radius 1 centered at origin
Math 45 Homework et olutions Points. ( pts) The integral is, x + z y d = x + + z da 8 6 6 where is = x + z 8 x + z = 4 o, is the disk of radius centered on the origin. onverting to polar coordinates then
More informationMath 461 Homework 8. Paul Hacking. November 27, 2018
Math 461 Homework 8 Paul Hacking November 27, 2018 (1) Let S 2 = {(x, y, z) x 2 + y 2 + z 2 = 1} R 3 be the sphere with center the origin and radius 1. Let N = (0, 0, 1) S 2 be the north pole. Let F :
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 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 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 informationMATH 261 FINAL EXAM PRACTICE PROBLEMS
MATH 261 FINAL EXAM PRACTICE PROBLEMS These practice problems are pulled from the final exams in previous semesters. The 2-hour final exam typically has 8-9 problems on it, with 4-5 coming from the post-exam
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 informationMath 20E Midterm II(ver. a)
Name: olutions tudent ID No.: Discussion ection: Math 20E Midterm IIver. a) Fall 2018 Problem core 1 /24 2 /25 3 /26 4 /25 Total /100 1. 24 Points.) Consider the force field F 5y ı + 3y 2 j. Compute the
More informationMath 461 Homework 8 Paul Hacking November 27, 2018
(1) Let Math 461 Homework 8 Paul Hacking November 27, 2018 S 2 = {(x, y, z) x 2 +y 2 +z 2 = 1} R 3 be the sphere with center the origin and radius 1. Let N = (0, 0, 1) S 2 be the north pole. Let F : S
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 informationMath 3c Solutions: Exam 1 Fall Graph by eliiminating the parameter; be sure to write the equation you get when you eliminate the parameter.
Math c Solutions: Exam 1 Fall 16 1. Graph by eliiminating the parameter; be sure to write the equation you get when you eliminate the parameter. x tan t x tan t y sec t y sec t t π 4 To eliminate the parameter,
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 informationPart I: Multiple Choice Mark the correct answer on the bubble sheet provided. n=1. a) None b) 1 c) 2 d) 3 e) 1, 2 f) 1, 3 g) 2, 3 h) 1, 2, 3
Math (Calculus II) Final Eam Form A Fall 22 RED KEY Part I: Multiple Choice Mark the correct answer on the bubble sheet provided.. Which of the following series converge absolutely? ) ( ) n 2) n 2 n (
More informationMath 230 Mock Final Exam Detailed Solution
Name: Math 30 Mock Final Exam Detailed Solution Disclaimer: This mock exam is for practice purposes only. No graphing calulators TI-89 is allowed on this test. Be sure that all of your work is shown and
More informationMAC2313 Final A. (5 pts) 1. How many of the following are necessarily true? i. The vector field F = 2x + 3y, 3x 5y is conservative.
MAC2313 Final A (5 pts) 1. How many of the following are necessarily true? i. The vector field F = 2x + 3y, 3x 5y is conservative. ii. The vector field F = 5(x 2 + y 2 ) 3/2 x, y is radial. iii. All constant
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 16, Advanced Calculus for Engineers
McGill University April 16, 2014 Faculty of cience Final examination Advanced Calculus for Engineers Math 264 April 16, 2014 Time: 6PM-9PM Examiner: Prof. R. Choksi Associate Examiner: Prof. A. Hundemer
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 information(x 3)(x + 5) = (x 3)(x 1) = x + 5
RMT 3 Calculus Test olutions February, 3. Answer: olution: Note that + 5 + 3. Answer: 3 3) + 5) = 3) ) = + 5. + 5 3 = 3 + 5 3 =. olution: We have that f) = b and f ) = ) + b = b + 8. etting these equal
More informationIntegrals. D. DeTurck. January 1, University of Pennsylvania. D. DeTurck Math A: Integrals 1 / 61
Integrals D. DeTurck University of Pennsylvania January 1, 2018 D. DeTurck Math 104 002 2018A: Integrals 1 / 61 Integrals Start with dx this means a little bit of x or a little change in x If we add up
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 informationEXAM. Practice for Second Exam. Math , Fall Nov 4, 2003 ANSWERS
EXAM Practice for Second Eam Math 135-006, Fall 003 Nov 4, 003 ANSWERS i Problem 1. In each part, find the integral. A. d (4 ) 3/ Make the substitution sin(θ). d cos(θ) dθ. We also have Then, we have d/dθ
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 information2016 FAMAT Convention Mu Integration 1 = 80 0 = 80. dx 1 + x 2 = arctan x] k2
6 FAMAT Convention Mu Integration. A. 3 3 7 6 6 3 ] 3 6 6 3. B. For quadratic functions, Simpson s Rule is eact. Thus, 3. D.. B. lim 5 3 + ) 3 + ] 5 8 8 cot θ) dθ csc θ ) dθ cot θ θ + C n k n + k n lim
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 informationDO NOT BEGIN THIS TEST UNTIL INSTRUCTED TO START
Math 265 Student name: KEY Final Exam Fall 23 Instructor & Section: This test is closed book and closed notes. A (graphing) calculator is allowed for this test but cannot also be a communication device
More informationMath 265 (Butler) Practice Midterm III B (Solutions)
Math 265 (Butler) Practice Midterm III B (Solutions). Set up (but do not evaluate) an integral for the surface area of the surface f(x, y) x 2 y y over the region x, y 4. We have that the surface are is
More informationa k 0, then k + 1 = 2 lim 1 + 1
Math 7 - Midterm - Form A - Page From the desk of C. Davis Buenger. https://people.math.osu.edu/buenger.8/ Problem a) [3 pts] If lim a k = then a k converges. False: The divergence test states that if
More informationMATH2321, Calculus III for Science and Engineering, Fall Name (Printed) Print your name, the date, and then sign the exam on the line
MATH2321, Calculus III for Science and Engineering, Fall 218 1 Exam 2 Name (Printed) Date Signature Instructions STOP. above. Print your name, the date, and then sign the exam on the line This exam consists
More informationFriday 09/15/2017 Midterm I 50 minutes
Fa 17: MATH 2924 040 Differential and Integral Calculus II Noel Brady Friday 09/15/2017 Midterm I 50 minutes Name: Student ID: Instructions. 1. Attempt all questions. 2. Do not write on back of exam sheets.
More information(x 3)(x + 5) = (x 3)(x 1) = x + 5. sin 2 x e ax bx 1 = 1 2. lim
SMT Calculus Test Solutions February, x + x 5 Compute x x x + Answer: Solution: Note that x + x 5 x x + x )x + 5) = x )x ) = x + 5 x x + 5 Then x x = + 5 = Compute all real values of b such that, for fx)
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 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 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 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 informationMath 234 Final Exam (with answers) Spring 2017
Math 234 Final Exam (with answers) pring 217 1. onsider the points A = (1, 2, 3), B = (1, 2, 2), and = (2, 1, 4). (a) [6 points] Find the area of the triangle formed by A, B, and. olution: One way to solve
More information1993 AP Calculus AB: Section I
99 AP Calculus AB: Section I 90 Minutes Scientific Calculator Notes: () The eact numerical value of the correct answer does not always appear among the choices given. When this happens, select from among
More informationThe answers below are not comprehensive and are meant to indicate the correct way to solve the problem. sin
Math : Practice Final Answer Key Name: The answers below are not comprehensive and are meant to indicate the correct way to solve the problem. Problem : Consider the definite integral I = 5 sin ( ) d.
More informationMidterm Exam #1. (y 2, y) (y + 2, y) (1, 1)
Math 5B Integral Calculus March 7, 7 Midterm Eam # Name: Answer Key David Arnold Instructions. points) This eam is open notes, open book. This includes any supplementary tets or online documents. You are
More informationWithout fully opening the exam, check that you have pages 1 through 10.
MTH 234 Solutions to Exam 2 April 11th 216 Name: Section: Recitation Instructor: INSTRUTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages 1 through
More informationPower Series. x n. Using the ratio test. n n + 1. x n+1 n 3. = lim x. lim n + 1. = 1 < x < 1. Then r = 1 and I = ( 1, 1) ( 1) n 1 x n.
.8 Power Series. n x n x n n Using the ratio test. lim x n+ n n + lim x n n + so r and I (, ). By the ratio test. n Then r and I (, ). n x < ( ) n x n < x < n lim x n+ n (n + ) x n lim xn n (n + ) x
More informationMath 263 Final. (b) The cross product is. i j k c. =< c 1, 1, 1 >
Math 63 Final Problem 1: [ points, 5 points to each part] Given the points P : (1, 1, 1), Q : (1,, ), R : (,, c 1), where c is a parameter, find (a) the vector equation of the line through P and Q. (b)
More informationMATH 31B: BONUS PROBLEMS
MATH 31B: BONUS PROBLEMS IAN COLEY LAST UPDATED: JUNE 8, 2017 7.1: 28, 38, 45. 1. Homework 1 7.2: 31, 33, 40. 7.3: 44, 52, 61, 71. Also, compute the derivative of x xx. 2. Homework 2 First, let me say
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 informationSection 6-5 : Stokes' Theorem
ection 6-5 : tokes' Theorem In this section we are going to take a look at a theorem that is a higher dimensional version of Green s Theorem. In Green s Theorem we related a line integral to a double 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 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 information( ) = x( u, v) i + y( u, v) j + z( u, v) k
Math 8 ection 16.6 urface Integrals The relationship between surface integrals and surface area is much the same as the relationship between line integrals and arc length. uppose f is a function of three
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 informationSection 5-7 : Green's Theorem
Section 5-7 : Green's Theorem In this section we are going to investigate the relationship between certain kinds of line integrals (on closed paths) and double integrals. Let s start off with a simple
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 informationGreen s Theorem. MATH 311, Calculus III. J. Robert Buchanan. Fall Department of Mathematics. J. Robert Buchanan Green s Theorem
Green s Theorem MATH 311, alculus III J. obert Buchanan Department of Mathematics Fall 2011 Main Idea Main idea: the line integral around a positively oriented, simple closed curve is related to a double
More information1. Evaluate the integrals. a. (9 pts) x e x/2 dx. Solution: Using integration by parts, let u = x du = dx and dv = e x/2 dx v = 2e x/2.
MATH 8 Test -SOLUTIONS Spring 4. Evaluate the integrals. a. (9 pts) e / Solution: Using integration y parts, let u = du = and dv = e / v = e /. Then e / = e / e / e / = e / + e / = e / 4e / + c MATH 8
More information36. Double Integration over Non-Rectangular Regions of Type II
36. Double Integration over Non-Rectangular Regions of Type II When establishing the bounds of a double integral, visualize an arrow initially in the positive x direction or the positive y direction. A
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 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 informationReview for the First Midterm Exam
Review for the First Midterm Exam Thomas Morrell 5 pm, Sunday, 4 April 9 B9 Van Vleck Hall For the purpose of creating questions for this review session, I did not make an effort to make any of the numbers
More informationy = x 3 and y = 2x 2 x. 2x 2 x = x 3 x 3 2x 2 + x = 0 x(x 2 2x + 1) = 0 x(x 1) 2 = 0 x = 0 and x = (x 3 (2x 2 x)) dx
Millersville University Name Answer Key Mathematics Department MATH 2, Calculus II, Final Examination May 4, 2, 8:AM-:AM Please answer the following questions. Your answers will be evaluated on their correctness,
More informationMATH 2433 Homework 1
MATH 433 Homework 1 1. The sequence (a i ) is defined recursively by a 1 = 4 a i+1 = 3a i find a closed formula for a i in terms of i.. In class we showed that the Fibonacci sequence (a i ) defined by
More informationParametric Equations and Polar Coordinates
Parametric Equations and Polar Coordinates Parametrizations of Plane Curves In previous chapters, we have studied curves as the graphs of functions or equations involving the two variables x and y. Another
More informationSummary of various integrals
ummary of various integrals Here s an arbitrary compilation of information about integrals Moisés made on a cold ecember night. 1 General things o not mix scalars and vectors! In particular ome integrals
More informationMath 102 Spring 2008: Solutions: HW #3 Instructor: Fei Xu
Math Spring 8: Solutions: HW #3 Instructor: Fei Xu. section 7., #8 Evaluate + 3 d. + We ll solve using partial fractions. If we assume 3 A + B + C, clearing denominators gives us A A + B B + C +. Then
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 informationSpring 2011 solutions. We solve this via integration by parts with u = x 2 du = 2xdx. This is another integration by parts with u = x du = dx and
Math - 8 Rahman Final Eam Practice Problems () We use disks to solve this, Spring solutions V π (e ) d π e d. We solve this via integration by parts with u du d and dv e d v e /, V π e π e d. This is another
More information3. On the grid below, sketch and label graphs of the following functions: y = sin x, y = cos x, and y = sin(x π/2). π/2 π 3π/2 2π 5π/2
AP Physics C Calculus C.1 Name Trigonometric Functions 1. Consider the right triangle to the right. In terms of a, b, and c, write the expressions for the following: c a sin θ = cos θ = tan θ =. Using
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 informationMATH 18.01, FALL PROBLEM SET # 6 SOLUTIONS
MATH 181, FALL 17 - PROBLEM SET # 6 SOLUTIONS Part II (5 points) 1 (Thurs, Oct 6; Second Fundamental Theorem; + + + + + = 16 points) Let sinc(x) denote the sinc function { 1 if x =, sinc(x) = sin x if
More informationMATH 1080 Test 2 -Version A-SOLUTIONS Fall a. (8 pts) Find the exact length of the curve on the given interval.
MATH 8 Test -Version A-SOLUTIONS Fall 4. Consider the curve defined by y = ln( sec x), x. a. (8 pts) Find the exact length of the curve on the given interval. sec x tan x = = tan x sec x L = + tan x =
More informationSOME PROBLEMS YOU SHOULD BE ABLE TO DO
OME PROBLEM YOU HOULD BE ABLE TO DO I ve attempted to make a list of the main calculations you should be ready for on the exam, and included a handful of the more important formulas. There are no examples
More informationMath Calculus II Homework # Due Date Solutions
Math 35 - Calculus II Homework # - 007.08.3 Due Date - 007.09.07 Solutions Part : Problems from sections 7.3 and 7.4. Section 7.3: 9. + d We will use the substitution cot(θ, d csc (θ. This gives + + cot
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 101 Midterm Examination Spring 2009
MATH Midterm Eamination Spring 9 Date: May 5, 9 Time: 7 minutes Surname: (Please, print!) Given name(s): Signature: Instructions. This is a closed book eam: No books, no notes, no calculators are allowed!.
More informationArc Length and Surface Area in Parametric Equations
Arc Length and Surface Area in Parametric Equations MATH 211, Calculus II J. Robert Buchanan Department of Mathematics Spring 2011 Background We have developed definite integral formulas for arc length
More informationPRELIM 2 REVIEW QUESTIONS Math 1910 Section 205/209
PRELIM 2 REVIEW QUESTIONS Math 9 Section 25/29 () Calculate the following integrals. (a) (b) x 2 dx SOLUTION: This is just the area under a semicircle of radius, so π/2. sin 2 (x) cos (x) dx SOLUTION:
More informationCalculus 152 Take Home Test 2 (50 points)
Calculus 5 Take Home Test (5 points) Due Tuesday th November. The following test will be done at home in order to ensure that it is a fair and representative reflection of your own ability in mathematics
More information1 Exponential Functions Limit Derivative Integral... 5
Contents Eponential Functions 3. Limit................................................. 3. Derivative.............................................. 4.3 Integral................................................
More informationMaterial for review. By Lei. May, 2011
Material for review. By Lei. May, 20 You shouldn t only use this to do the review. Read your book and do the example problems. Do the problems in Midterms and homework once again to have a review. Some
More informationMAT 211 Final Exam. Spring Jennings. Show your work!
MAT 211 Final Exam. pring 215. Jennings. how your work! Hessian D = f xx f yy (f xy ) 2 (for optimization). Polar coordinates x = r cos(θ), y = r sin(θ), da = r dr dθ. ylindrical coordinates x = r cos(θ),
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 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 informationMidterm 1 practice UCLA: Math 32B, Winter 2017
Midterm 1 practice UCLA: Math 32B, Winter 2017 Instructor: Noah White Date: Version: practice This exam has 4 questions, for a total of 40 points. Please print your working and answers neatly. Write your
More information5.9 Representations of Functions as a Power Series
5.9 Representations of Functions as a Power Series Example 5.58. The following geometric series x n + x + x 2 + x 3 + x 4 +... will converge when < x
More informationMath 2E Selected Problems for the Final Aaron Chen Spring 2016
Math 2E elected Problems for the Final Aaron Chen pring 216 These are the problems out of the textbook that I listed as more theoretical. Here s also some study tips: 1) Make sure you know the definitions
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 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 informationChapter 9 Overview: Parametric and Polar Coordinates
Chapter 9 Overview: Parametric and Polar Coordinates As we saw briefly last year, there are axis systems other than the Cartesian System for graphing (vector coordinates, polar coordinates, rectangular
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 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 informationMATH 280 Multivariate Calculus Fall Integration over a curve
dr dr y MATH 28 Multivariate Calculus Fall 211 Integration over a curve Given a curve C in the plane or in space, we can (conceptually) break it into small pieces each of which has a length ds. In some
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 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 informationFall 2013 Hour Exam 2 11/08/13 Time Limit: 50 Minutes
Math 8 Fall Hour Exam /8/ Time Limit: 5 Minutes Name (Print): This exam contains 9 pages (including this cover page) and 7 problems. Check to see if any pages are missing. Enter all requested information
More information10.1 Review of Parametric Equations
10.1 Review of Parametric Equations Recall that often, instead of representing a curve using just x and y (called a Cartesian equation), it is more convenient to define x and y using parametric equations
More information