In-class quiz B solution guide

Problem: Set up, but do not evaluate, a double integral representing the solid bounded by the surfaces z=0, x-y=0, x=0, x+y=2 and z=y²+2.

The first step is to sketch the region and an appropriate domain of integration. Fortunately, all but one of the surfaces are planes, so it's not tough to sketch them all.

The surfaces are shown on the left, and the region of integration in the xy plane is shown on the right. You can use the parabola in yand z as the top of the volume and the z=0 plane as the bottom. There are other possibilities, but these are harder to set up. Now, you must decide upon an order of integration. The most direct approach is the integrate in the xdirection on the outside.

$$V = \int_0^1\int_{x}^{2-x} (y^2+2) dy dx$$

If you choose to integrate in the y direction on the outside, you will need two integrals:

$$V = \int_0^1\int_0^{y} (y^2+2) dx dy+ \int_1^2\int_{0}^{2-y} (y^2+2) dx dy.$$