Exam 14: Multiple Integrals

A function is a pdf on the three-dimensional region Q if for all in Q and . Find k such that is a pdf on the region bounded by z = 0, z = 3, x = 0, x = 2, y - z = 0 , and y - z = 3.

Find the volume of the given solid Q. Q is bounded by x - 4z = -1, x - 4z = 2, -2y + 3z = 3, -2y + 3z = 5, -4y + z = 1, and -4y + z = 2.

A skydiving club is having a competition to see who can land the closest to a target point. Jeff is a highly experienced skydiver, and the probability that he will land inside a region R is given by , where the coordinate system is centered on the target point. Compute the probability that Jeff lands within 11 feet of the target point.

Convert the point to rectangular coordinates (x,y,z).

Evaluate the iterated integral by first changing the order of integration.

Evaluate by converting to polar coordinates.

Find an integral equal to the volume of the solid bounded by the given surfaces and evaluate the integral.

Find a transformation from a rectangular region S in the uv-plane to the region R which is bounded by and

Rewrite the iterated integral by iterating in the order

Use an appropriate coordinate system to compute the volume of the solid below , above and inside .

Set up and evaluate the integral where Q is the region above the xy-plane bounded by the hemisphere centered at (0,0,0) and with a radius of 5.

Find the surface area of the portion of above .

Evaluate the iterated integral by first changing the order of integration.

Evaluate the triple integral . ,

Convert the equation into spherical coordinates.

Evaluate the integral , where Q is the region with z > 0 bounded by and .

Convert the point to rectangular coordinates (x,y,z).

Evaluate the iterated integral.

Find the surface area of the portion of above the xy plane.

Find the mass of the solid with density and the given shape. ,