Exam 14: Section 6: Multiple Integration

Find of the center of mass of the solid of given density bounded by the graphs of the equations .

Find the average value of over the region Q, where Q is a cube in the first octant bounded by the coordinate planes, and the planes and . The average value of a continuous function over a solid region Q is , where V is the volume of the solid region Q.

Use a triple integral to find the volume of the solid bounded by the graphs of the equations .

Evaluate the iterated integral .

Sketch the solid whose volume is given by the iterated integral given below and use the sketch to rewrite the integral using the indicated order of integration. Rewrite the integral using the order .

Determine the value of b so that the volume of the ellipsoid is .

Use a triple integral to find the volume of the solid shown below.

Find the center of mass of the solid bounded by and with density function .

Set up a triple integral that gives the moment of inertia about the -axis of the solid region Q of density given below.

Evaluate the following iterated integral.

Find the centroid of the solid region bounded by the graphs of the equations. Use a computer algebra system to evaluate the triple integral. (Assume uniform density and find the center of mass.)

Use a triple integral to find the volume of the solid shown below.

Set up a triple integral for the volume of the solid bounded above by the cylinder and below by the paraboloid .

Set up a triple integral for the volume of the solid bounded by and .

Find for the indicated solid with density function .

Evaluate the following iterated integral.

Set up a triple integral for the volume of the solid bounded by the coordinate planes and the plane given below.

Find the average value of over the region Q, where Q is a tetrahedron in the first octant with vertices and . The average value of a continuous function over a solid region Q is , where V is the volume of the solid region Q.

Rewrite the iterated integral using the order .