We'll hand out more problems at the exam review on Monday, May 7th. Here are a few to get you started. These should give you an idea of what we're interested in, and you can study related problems from the Chapter Reviews and sections in your book.

Remember what we said in class: when students take the homework less seriously (which has happened this semester) and especially if we have to talk about specific sections in class (which Eoin did for section 15.7), we occasionally take a problem or two directly from homework or groupwork and put it on the test.

Since we spent a full day on drawing and sketching, you can expect to have to sketch a number of different solids, etc., on the exam.

Also, the problems here don't (yet) include anything on general change of variables for triple integrals and curl/divergence, but those topics will appear on the test. I'll add some related problems to this document over the weekend and/or at the review session on Monday, but you can start studying problems in 15.9 (the 3D part) and 16.5.

Note: The problems below have been given on final exams in math department courses during the last few years. A few of them problems on your exam will have roughly the same level of difficulty. A few others could will be more "UMTYMP-like" and require you do so some geometric reasoning, justifying your answers, and so on.

1. Let B be the solid bounded by the cylinder x²+y² = 9 and the planes z = 0 and z= 2. Let F be the vector field = F(x,y,z) = (x, y, (4-z²)(x²+y²)) and let f = Div F. Thus f(x,y,z) = 2-2z(x²+y²). Calculate the integral of f over the solid region B.

2. Let S be the surface which is the boundary of the solid B used in problem 1. Let F be the vector field mentioned in problem 1. That is, F(x,y,z) = (x, y, (4-z²)(x²+y²)). Find the integral of F over S by direct calculation. Use the outward pointing normal. Don't use any "big" theorem, such as Green's Theorem, Stokes's Theorem, or the Divergence Theorem.

3. Let P be the value of the integral in problem 1. Let Q be the value of the integral in problem 2. Is P greater than Q, is P less than Q, or is P equal to Q? Explain why this happens.

4. Let S be the solid bounded above by the sphere x² + y² + z² = 16 and below by the paraboloid z = (x² + y²)/6.

   (a) Parametrize the solid S using cylindrical coordinates.

   (b) Let F(x, y, z) = (x + yz, sin(x⁹z⁶), cos(x⁷y⁸)). Use the Divergence Theorem to find the flux of F through the boundary of S. Use the outward pointing normal.

5. Let F(x, y, z) = (- y, x, sin(x⁷y⁸z⁹)) and let M be the surface given by z = x² + y² - 9, z < 0.

   (a) Give a parametrization of the boundary of M as a curve in 3-space.

   (b) Using any valid method, evaluate the integral of curl F over the surface M, where M has the outward pointing normal.

6. Rewrite the integral as an integral where the order of integration is dydzdx. HINT (and a way to get partial credit): For each fixed value of x in the range 1 < x < 2, draw the region of integration in the (y,z) plane.
7. Let F(x,y,z) = so that curl F =

   Let S be the hemisphere given by , z > 0. Using spherical coordinates, set up, but do not evaluate, an integral which represents the flux of curl F across S, using the outward pointing normal. Your answer should be expressed as the integral of some function, with appropriate endpoints of integration. Your answer should not contain any dot products, cross products, etc. These should all be worked out and simplified. Then use Stokes's Theorem to evaluate the integral.