91Ó°ÊÓ

The following notation is used in the problems: \(M=\) mass, \(\bar{x}, \bar{y}, \bar{z}=\) coordinates of center of mass (or centroid if the density is constant), \(I=\) moment of inertia (about axis stated), \(I_{x}, I_{y}, I_{z}=\) moments of inertia about \(x, y, z\) axes, \(I_{m}=\) moment of inertia (about axis stated) through the center of mass. Note: It is customary to give answers for \(I, I_{m}, I_{x},\) etc., as multiples of \(M\) (for example, \(I=\frac{1}{3} M l^{2}\) ). Prove the "parallel axis theorem": The moment of inertia I of a body about a given axis is \(I=I_{m}+M d^{2},\) where \(M\) is the mass of the body, \(I_{m}\) is the moment of inertia of the body about an axis through the center of mass and parallel to the given axis, and \(d\) is the distance between the two axes.

(a) Write a triple integral in cylindrical coordinates for the volume of the part of a ball between two parallel planes which intersect the ball. (b) Evaluate the integral in (a). Warning hint: Do the \(r\) and \(\theta\) integrals first. (c) Find the centroid of this volume.

Find the volume in the first octant bounded by the coordinate planes and the plane \(x+2 y+z=4\).