Calculus II, Spring 2013

Final Exam

June 20, 2013, 10:10 AM. Show all details.

1. (10 pts) Let D = {1 ≤ x ≤ 2, 0 ≤ xy ≤ 1, 0 ≤ z ≤ 1}. Evaluate

∫ ∫ ∫_{D}(x^{2}y + xyz)dV.

2. (10 pts) Find the work done by F (x, y, z) = (xy, yz, x^{2}) over the line segment joining \underline{from} (2, 2, 2) \underline{to} (1, 1, 1).

3. (10 pts) Integrate g(x, y, z) = 2x − y + z over the portion of the plane x + y + z = 1 that lies in the first octant.

4. (15 pts) Let R = {1 ≤ 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 ∫ F · dr, ∫ G · dr and ∫ H · dr is (are) zero on \underline{every} closed loop in R?) Explain.

5. (40 pts) Let D = {x^{2} + y^{2} + z^{2} < 4, x > 0}, S = {x^{2} + y^{2} + z^{2} = 1, z > 0} and F (x, y, z) = (x + z, xz, xy).

(a) Which one(s) of S and D does Stokes Theorem apply? Verify Stokes Theorem on it (them). That is, compute integrals on both sides of the Stokes Theorem and check they are the same.

(b) Do the same for divergence Theorem.

6. (15 pts) Let F (x, y, z) = \frac{(x, y, z)}{(\sqrt{x^{2} + y^{2} + z^{2}})^{3}} . Evaluate _{∫ ∫\frac{(x−1)^{2}}{4} +y^{2}+z^{2}=1}F · n dσ.