MULTIVARIATE CALCULUS, CH 15 - STUDY GUIDE
1. Set up the double integral for the volume under the surface z = _______ and over the rectangle R={(x,y): 0£x£___, 0£y£___}. Evaluate the integral as far as practical manually, and finish the integration with a calculator.
2. Use a double integral in polar coordinates to find the area of the region inside one loop of the function r = _________.
3. Evaluate 
.
4. Evaluate a double integral by first reversing the order of integration.
Example: 
5. Find the surface area of that portion of the function z = _________ which lies above the triangular region with vertices at (0,0,0), (2,0,0) and (2,1,0) [or some other description of the region may be given].
6. Find the surface area for that portion of a parameterized surface r(u,v) = <x(u,v),y(u,v),z(u,v)> for -2<u<2, -3<v<3
7. Find an equation of the tangent plane at a given point for a surface given parametrically
8. Find the volume and centroid of the solid enclosed by the paraboloid z=2x2+2y2 and the plane z=____ (hint: translate to cylindrical coordinates).
9. Use spherical coordinates to find the mass of the "ice cream cone" shaped solid below the sphere of radius ___ and above the cone z = ______ if its density is inversely proportional to the distance from the center.
10. Rewrite the integral using an appropriate change of variables to
simplify the region: 
where R is the region bounded by the parallelogram with
vertices (0,0), (-2,3), (2,5), and
(4,2). Alternately, you may be asked to
simplify the integrand by an appropriate change of variables. I may of may not ask you to evaluate the
resulting integral.
Return to: Merced College; Don Power Updated 04/28/04 by Don Power