An example

Set up an iterated integral to compute the volume of a region that is bounded by (a) the xy-plane, (b) the cone $z = \sqrt{x^2 + y^2}$, and (c) the cylinder x² + y² = 4, with a density function $\delta(x,y,z) = z$.

1.

First, draw the region. It's the shape you'd get if you removed a conical piece from the center of a cylinder. (illus of region)

2.

Find the limits of integration. Let's integrate in the order: z,y,x. This gives us: $0 \le z \le \sqrt{x^2 + y^2}, -\sqrt{4 - x^2} \le y \le \sqrt{4-x^2}, -2 \le x \le 2$.

3.

Set up the integral and integrate:

$$\int_W \delta dV = \int_{-2}^2 \int_{-\sqrt{4-x^2}}^{\sqrt{4-x^2}} \int_0^{\sqrt{x^2 + y^2}} z dz dy dx$$

$$= \int_{-2}^2 \int_{-\sqrt{4-x^2}}^{\sqrt{4-x^2}} \left[\frac{1}{2} z^2 \right]_0^{\sqrt{x^2+y^2}} dy dx$$

$$= \int_{-2}^2 \int_{-\sqrt{4-x^2}}^{\sqrt{4-x^2}} \frac{1}{2}(x^2 + y^2) dy dx$$

$$= \frac{1}{2} \int_{-2}^2 \left[x^2y + \frac{1}{3}y^3\right]_{-\sqrt{4-x^2}}^{\sqrt{4-x^2}} dx$$

$$= \frac{2}{3} \int_{-2}^2 (2 + x^2)\sqrt{4 - x^2} dx = \frac{160}{9}.$$