Calculus II, Spring 2014

Final Exam

June 19, 2014, 10:10 AM. Show all details.

1. (15 pts) Replace

∫_{0}^{2π} ∫_{0}^{1} ∫_{0}^{√\overline{2−r^{2}}} rdz dr dθ

by triple integrals in spherical coordinates in the order dρdϕdθ and dϕdρdθ, respectively. Need not evaluate them.

2. (15 pts) Let D = {1 ≤ x ≤ 2, 0 ≤ xy ≤ 1, 0 ≤ z ≤ 1}. Evaluate

∫ ∫ ∫_{D}(x^{2}y + xyz)dV.

3. Let F (x, y) = (M(x, y), N(x, y)) have continuous first and second derivatives everywhere in ℝ^{2} and let R = {x^{2} + y^{2} < 1, x > 0, y > 0}.

(a) (10 pts) State Green’s Theorem in both forms for F on R.

(b) Prove either one of the two forms on R (20 pts). (10 pts) If you can not prove it for general F, do the following: Take F = (y, x) and verify both forms on R. That is, evaluate both line integral and double integral and check that they are the same. Do this for both forms.

4. (15 pts) True or false? Give details.

If f(x, y, z) has continuous first derivatives in a domain D, and C = {(x(t), y(t), z(t)), 0 ≤ t ≤ 1} be a smooth curve in D. Then ∫_{C} ∇f · T ds depends only on f, (x(0), y(0), z(0)) and (x(1), y(1), z(1)).

5. (30 pts) Let R = {1/4 ≤ x^{2} + y^{2} ≤ 4}, F (x, y) = (2y, x), G(x, y) = (x, y), H(x, y) =

(\frac{−y}{x^{2} + y^{2}} , \frac{x}{x^{2} + y^{2}} ).

Which one(s) of F, G and H is (are) conservative on R? (That is, which one(s) of ∫_{C} F · dr, ∫_{C} G · dr and ∫_{C} H · dr is (are) zero on \underline{every} closed loop C in R?) Explain.

6. (15 pts) Let F (x, y, z) = zk and S = {x^{2} + y^{2} + z^{2} = 4, x > 0, y > 0, z > 0}.

Evaluate _{∫ ∫S}F · n dσ where n is unit normal of S pointing away from the origin.