An example

The center of mass of an object, W, with a density function $\delta$ has coordinates (x_c,y_c,z_c) which are given by

$$x_c = \frac{1}{\mbox{mass}}\int_W x\delta dV, \quad y_c = \f... ...y\delta dV, \quad z_c = \frac{1}{\mbox{mass}}\int_W z\delta dV.$$

Find the center of mass of a cylindrical wedge with $\delta (x,y,z) = 1$,where the wedge W is given by $0 \le r \le 4, 0 \le \theta \le \pi/4, 0 \le z \le 2$.

1.

Calculate the total mass:

$$\mbox{mass} = \int_W \delta(x,y,z) dV$$

$$= \int_0^4 \int_0^{\pi/4} \int_0^2 1 r dz d\theta dr$$

$$= \int_0^4 \int_0^{\pi/4} r \left. z \right\vert _0^2 d\theta dr$$

$$= \int_0^4 \int_0^{\pi/4} 2r d\theta dr$$

$$= \int_0^4 (2r) \left. \theta \right\vert _0^{\pi/4} dr$$

$$= \int_0^4 \frac{\pi}{2} r dr = 4\pi.$$

2.

Next calculate x_c: (Note $x = r\cos \theta$)

$$x_c = \frac{1}{4\pi} \int_W x\delta dV = \frac{1}{4\pi} \int... ..._0^4 (r \cos \theta) r dr d\theta dz = \frac{16\sqrt{2}}{3\pi}.$$

3.

Calculate y_c using $y = r\sin \theta$">:

$$y_c = \frac{1}{4\pi} \int_W y\delta dV = \frac{1}{4\pi} \int... ...(r\sin \theta) r dr d\theta dz = \frac{16(2 - \sqrt{2})}{3\pi}.$$

4.

Calculate z_c: (note z is the same in both coordinates)

$$z_c = \frac{1}{4\pi} \int_W z\delta dV = \frac{1}{4\pi} \int_0^2 \int_0^{\pi/2} \int_0^4 zr dr d\theta dz = 1.$$