Calculus II, Spring 2009

Final Exam

June 25, 2009, 10:10 AM. Show all details.

1. Let R be the region in the first quadrant bounded by xy = 1, xy = 4 and the lines y = x, y = \frac{x}{4}. Compute ∫ ∫_{R}(\sqrt{\frac{y}{x}} + \sqrt{xy})dxdy.

2. 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. 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. 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, ∫ F · dr and ∫ F · dr is (are) zero on \underline{every} closed loop in R?)

5. Let F (x, y) = (M(x, y), N(x, y)) where both M and N are continuous together with their first derivatives with in a region R ∈ ℝ^{2}. State (need not prove) Green’s Theorem in normal and tangential forms.

6. Let D = {x^{2} + y^{2} + z^{2} ≤ 1, z ≥ 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.