Math 52 - Fall Final Exam PART 1

Transcription

1 Math 52 - Fall Final Exam PART 1 Name: Student ID: Signature: Instructions: Print your name and student ID number and write your signature to indicate that you accept the Honor Code. This exam consists of two parts separated for the ease of grading. Please sign booth parts of the exam. There are 10 problems total: 5 in the first part on the pages numbered from 1 to 5, and other 5 problems on pages 6 to 10 stapled separately. There is a total of 100 points. Point values are given in parentheses. Please check that the version of the exam you have is complete, and correctly stapled. Read each question carefully. In order to receive full credit, please show all of your work and justify your answers. You do not need to simplify your answers unless specifically instructed to do so. You may use any result from class that you like, but if you cite a theorem be sure to verify the hypotheses are satisfied. You have 3 hours. This is a closed-book, closed-notes exam. No calculators or other electronic aids will be permitted. If you finish early, you must hand your exam paper to a member of teaching staff. If you need extra room, use the back sides of each page. If you must use extra paper, make sure to write your name on it and attach it to this exam. Do not unstaple or detach pages from this exam.

2 Problem 1. (10 pts.) Compute the integral y e x3 dx dy 1

3 Problem 2. (10 pts.) Change the order of integration in the integral: 1 1 x+y f(x, y, z) dz dy dx into: f(x, y, z) dx dz dy 2

4 Problem 3. (10 pts.) Show that the area of the region D = { (x, y) R 1 xy 5 and 1 y x 2 and x > 0 } is equal to ln 4. 3

5 Problem 4. (10 pts.) Let C be the straight interval from (1, 0) to (2, 2). a) Compute xy ds C b) Compute C xy dy 4

6 Problem 5. (10 pts.) let Let C be the upper half of the circle x 2 + y 2 = 4 with y > 0 and ( F = 7x + e y2) i + (2 3y) j Compute the flux C F n ds where n is the unit normal vector pointing away from the center of the circle. Hint: Complete C to a closed path and use divergence form of the Green s Theorem. 5

7 The following boxes are strictly for grading purposes. Please do not mark. Question Score Maximum Total 50

8 Math 52 - Fall Final Exam PART 2 Name: Student ID: Signature: Instructions: Print your name and student ID number and write your signature to indicate that you accept the Honor Code. This exam consists of two parts separated for the ease of grading. Please sign booth parts of the exam. There are 10 problems total: 5 in the first part on the pages numbered from 1 to 5, and other 5 problems on pages 6 to 10 stapled separately. There is a total of 100 points. Point values are given in parentheses. Please check that the version of the exam you have is complete, and correctly stapled. Read each question carefully. In order to receive full credit, please show all of your work and justify your answers. You do not need to simplify your answers unless specifically instructed to do so. You may use any result from class that you like, but if you cite a theorem be sure to verify the hypotheses are satisfied. You have 3 hours. This is a closed-book, closed-notes exam. No calculators or other electronic aids will be permitted. If you finish early, you must hand your exam paper to a member of teaching staff. If you need extra room, use the back sides of each page. If you must use extra paper, make sure to write your name on it and attach it to this exam. Do not unstaple or detach pages from this exam.

9 Problem 6. (10 pts.) field F for which Let S be any closed oriented surface. Show that there is no vector F = n S where n S is the unit vector field orienting S. (Recall: closed surface means a surface with no boundary.) Hint: use Stokes theorem. 6

10 Problem 7. (10 pts.) the cylinder x 2 + y 2 = 4. Show that Let S be the surface cut from the set { (x, y, z) z = x 2 y 2 } by area (S) = π 6 ( 17 3/2 1 ) 7

11 Problem 8. (10 pts.) Let f(x, y) = ln (2x 2 + 5y 2 ). a) Compute (f(x, y) ) and indicate its domain. b) Let C be part of the parabola y = 9 x 2 over 3 x 2 oriented from left to right. Compute ( ln ( 2x 2 + 5y 2) ) T ds C 8

12 Problem 9. (10 pts.) Let S be the the cone z 2 = x 2 + y 2 for 0 z h, oriented with normal vector pointing up. (Note: this is just the lateral side of the cone.) Let F = x 2 i +y 2 j +3z k. Show that: S F n S ds = 2πh 3 Hint: add the top disk z = h, x 2 + y 2 h 2 on the cone and use divergence theorem. The volume of the solid cone is 1 3 πr2 h. 9

13 Problem 10. (10 pts.) Let C be the interval in the y = 0 plane from (1, 0, 1) to (2, 0, 2) and let S be the surface obtained by rotating the curve C about the x axis. (a) Parametrize the surface S. Write formulas for (x, y, z) coordinates in terms of your parameters and write the domain of your parameterization. (b) Orient S with the normal vector n S pointing away from the x axis, i.e. ns i < 0. Determine for what order of the parameters your parameterization will preserve the orientation of S. (c) Compute S x + z ds 10

14 The following boxes are strictly for grading purposes. Please do not mark. Question Score Maximum Total 50