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Math 215 – Calculus III Exam 2 11/14/
I Your PRINTED name is:
I Please circle your section and write down your lab time:
Section Time Professor GSI Your lab time 10 8-9 Dabkowski Kaye 20 8-9 Esedoglu Ngo 30 9-10 Esedoglu Zhao 40 10-11 D’Souza Ma 50 11-12 D’Souza Henry 60 12-1 D’Souza Zhou 70 1-2 Ruan Lee 80 2-3 Ruan Su 90 3-4 Uribe Froehlich
Instructions
I This is a 90 minutes closed book exam. No notes or calculators are allowed. I This examination booklet contains 5 problems, on 13 sheets of paper including this front cover. Check this before you start. I The next to last page is a list of formulas which you might find useful. The last page is blank and is to be used as scratch paper. Feel free to tear these last two pages away from the rest of the exam, but be careful to tear only those pages. I SHOW ALL YOUR WORK. Make sure your final answer is clearly written in the corresponding answer box. I DO NOT CHEAT! IF YOU CHEAT, YOU FAIL!
Grading
Total:
(d) [Multiple choice] Which of the following vector fields is conservative? Circle your answer : A B C
A B C
(e) [Multiple choice] Which of the following vector fields is the gradient vector field of f (x, y) = x^2 − 3 y^2? Circle your answer : A B C
A B C
2 (15 pts.) Consider the double integral
I =
0
∫ (^9) −x 2 0
xe^2 y 9 − y dydx. (a) Sketch the region of integration.
(b) Evaluate the integral by reversing the order of integration.
YOUR ANSWER:
The integral equals
(b) Find the centroid of E. (Hint: You don’t need to evaluate any extra integral for this problem.)
YOUR ANSWER:
The centroid of E is the point
4 (30 pts.) Consider the planar vector field F~ = 〈 3 x^2 y^2 − x, 2 x^3 y + 2y〉.
(a) Show that F~ is a conservative vector field.
(b) Find a potential function of F~.
YOUR ANSWER:
A potential function of F~ is
5 (20 pts.) Set up but do not evaluate the integrals necessary to find the following quantities. (Your integrals should contain explicit upper/lower bounds.) (a) The volume of the tetrahedron enclosed by the coordinate planes and the plane x + 3y + 4z = 12.
YOUR ANSWER:
The integral to be evaluated is
(b) The y-coordinate of the center of mass of a triangular lamina with vertices (1, 0), (1, 2), (2, 0), if the density function is equal to ρ(x, y) = 1 + x + y. [You need to write down two integrals.]
YOUR ANSWER:
The integrals to be evaluated are
(c) The mass of a wire in the shape of the helix x = t, y = cos t, z = sin t, 0 ≤ t ≤ 4 π, if the density at any point is equal to the square of the distance from the origin.
YOUR ANSWER:
The integral to be evaluated is
(d) The work done by the force field F~ (x, y) = 〈x^2 −z, − 2 xy, xz〉 in moving a particle along the curve ~r(t) = 〈t + 1, sin t, t^2 〉, 0 ≤ t ≤ 3.
YOUR ANSWER:
The integral to be evaluated is
This page is intended for use as scratch paper.
