An example

Compute the mass of the metal plate shown, if its density function is $\delta(x,y) = \sqrt{x^2 + y^2}$. Note that since the plate covers a region shaped like a portion of an annulus, polar coordinates are a good choice for integration.

1.

In polar coordinates, the density function becomes $\delta(x,y) = \sqrt{x^2 + y^2} \rightarrow f(r,\theta) = r$.

2.

The region of integration is $2 \le r \le 3, \frac{\pi}{4} \le \theta \le \frac{4\pi}{3}$.

3.

The mass is then

$$\int_R f(x,y) dA = \int_{\pi/4}^{4\pi/3} \int_2^3 f(r,\theta) r dr d\theta$$

$$= \int_{\pi/4}^{4\pi/3} \int_2^3 r^2 dr d\theta$$

$$= \int_{\pi/4}^{4\pi/3} \left.\frac{1}{3}r^3\right\vert _2^3 d\theta$$

$$= \int_{\pi/4}^{4\pi/3} 6 d\theta$$

$$= 6 \left. \theta \right\vert _{\pi/4}^{4\pi/3}$$

$$= \frac{13 \pi}{2}.$$