Calculus II, Spring 2005

Final Exam

June 16, 2004, 10:10 AM. \underline{SHOW ALL YOUR WORK}.

1. (15 pts) Use the method of Lagrangian multipliers (and NOT any other methods) to find the extreme values of the function f(x, y, z) = xy + z^{2} on the circle in which the plane y − 2x = 0 intersects the sphere x^{2} + y^{2} + z^{2} = 1.

2. (10 pts) Switch the order of integration of

∫_{0}^{1} ∫_{−1}^{0} ∫_{0}^{y^{2}} dzdydx

to dydxdz and dxdydz respectively. You don’t need to find the numerical value of the integral.

3. (15 pts) 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.

4. (15 pts) Find the integral of G(x, y, z) = x over the surface given by {y = x^{2}, 0 ≤ x ≤ 2, 0 ≤ z ≤ 3}

5. (15 pts) Let C be the upper half of the ellipse \frac{x^{2}}{4} + y^{2} = 1, y ≥ 0, counterclockwise oriented. F (x, y) = (−y, 3x). Compute ∫_{C} F · T ds.

6. (30 pts) Let D = {x^{2} + y^{2} + z^{2} ≤ 1, z ≥ 0}, S = {x^{2} + y^{2} + z^{2} = 1, z ≥ 0}. Verify the Stokes Theorem and divergence Theorem for F = (x+z, xz, xy) on S and D (Compute integrals on both sides and check they are the same).