Exercise 21

In each of the following questions, set up an iterated integral expression whose value determines the desired result. Then, evaluate the integral first by hand, and then using appropriate technology.

1. Find the volume of the “cap” cut from the solid sphere $x^2 + y^2 + z^2 = 4$ by the plane $z=1$, as well as the $z$-coordinate of its centroid.

2. Find the $x$-coordinate of the center of mass of the portion of the unit sphere that lies in the first octant (i.e., where $x$, $y$, and $z$ are all nonnegative). Assume that the density of the solid given by $\delta(x,y,z) = \frac{1}{1+x^2+y^2+z^2}$.

3. Find the volume of the solid bounded below by the $xy$-plane, on the sides by the sphere $\rho=2$, and above by the cone $\phi = \pi/3$.

4. Find the $z$ coordinate of the center of mass of the region that is bounded above by the surface $z = \sqrt{\sqrt{x^2 + y^2}}$, on the sides by the cylinder $x^2 + y^2 = 4$, and below by the $xy$-plane. Assume that the density of the solid is uniform and constant.

5. Find the volume of the solid that lies outside the sphere $x^2 + y^2 + z^2 = 1$ and inside the sphere $x^2 + y^2 + z^2 = 2z$.