91Ó°ÊÓ

(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$\mathit{θ}$integrals first.

(c) Find the centroid of this volume.

Find the moment of inertia about a diagonal of a framework consisting of the four sides of a square of side a.

Above the square with vertices at, (0,0), (2,0),(0,2) and (2,2) and under the plane z = 8-x+y.

Under the surface z = y(x+2) , and over the area bounded by $\mathbf{x}\mathbf{+}\mathbf{y}\mathbf{=}\mathbf{0}\mathbf{,}\mathbf{}\mathbf{y}\mathbf{=}\mathbf{1}\mathbf{,}\mathbf{}\mathbf{y}\mathbf{=}\sqrt{\mathbf{x}}$.

Use the parallel axis theorem (Problem 3.1)

(a) and Example 3, to find the moment of inertia of a solid ball about a line tangent to it;

(b) and Problem 3b to find the moment of inertia of a solid cylinder about a ruling

For a thin rod of length and uniform densityfind

(a) M,

(c)${\mathbf{l}}_{\mathbf{m}}$about an axis perpendicular to the rod,

(d)labout an axis perpendicular to the rod and passing through one end (see Problem 1).

Find the volume between the planes z = 2x + 3y +6 and z = 2x + 7y + 8, and over the triangle with vertices, (0,0) (3,0) and (2,1).

For a square lamina of uniform density, findabout

(a) a side,

(b) a diagonal,

(c) an axis through a corner and perpendicular to the plane of the lamina. Hint: See the perpendicular axis theorem, Example 1f.

A triangular lamina has vertices (0,0),(0,6) and(6,0),and uniform density. Find:

(a) $\overline{\mathbf{x}}\mathbf{,}\overline{\mathbf{y}}$

(b)${\mathit{I}}_{\mathbf{x}}$,

(c)${\mathit{I}}_{\mathbf{m}}$ about an axis parallel to thex-axis. Hint: Use Problemcarefully.

For a uniform cube, find Iabout one edge.