Exam 14: Section 7: Multiple Integration

Use cylindrical coordinates to find the volume of the solid inside both and .

Use cylindrical coordinates to find the volume of the solid bounded above by and below by .

Use cylindrical coordinates to find the volume of the cone where and .

Evaluate the following iterated integral.

Use spherical coordinates to find the z coordinate of the center of mass of the solid lying between two concentric hemispheres of radii 6 and 7, and having uniform density k.

Evaluate the following iterated integral.

Use spherical coordinates to find the mass of the sphere with the given density. The density at any point is proportional to the distance of the point from the z-axis.

Use spherical coordinates to find the z coordinate of the center of mass of the solid lying between two concentric hemispheres of radii 4 and 7, and having uniform density k.

Use spherical coordinates to find the volume of the solid inside and outside , and above the xy-plane.

Use spherical coordinates to find the volume of the solid inside the torus given by .

Evaluate the iterated integral .

Use spherical coordinates to find the mass of the sphere and density .

Convert the integral below from rectangular coordinates to both cylindrical and spherical coordinates, and evaluate the simpler iterated integral.

Convert the integral below from rectangular coordinates to both cylindrical and spherical coordinates, and evaluate the simpler iterated integral.

Find the "volume" of the "four-dimensional sphere" by evaluating .

Use spherical coordinates to find the volume of the solid between the spheres and and inside the cone .

Use cylindrical coordinates to find the volume of the solid inside the sphere and above the upper nappe of the cone .

Evaluate the following iterated integral.

Use cylindrical coordinates to find the mass of the solid where .