Exam 15: Multiple Integrals

Find the area of the surface S where S is the part of the plane that lies above the triangular region with vertices , and

Use cylindrical coordinates to evaluate the triple integral where E is the solid that lies between the sphere and in the first octant.

Use a computer algebra system to find the moment of inertia of the lamina that occupies the region D and has the density function , if .

The sketch of the solid is given below. Given , write the inequalities that describe it.

Use spherical coordinates to evaluate where B is the ball

Evaluate the integral where and with respect to x, y, and z, in that order.

Find the area of the part of hyperbolic paraboloid that lies between the cylinders and .

Evaluate the double integral , where is the region bounded by the graphs of and .

Evaluate where is the figure bounded by and .

Compute , where is the disk , by first identifying the integral as the volume of a solid.

Express the volume of the wedge in the first octant that is cut from the cylinder by the planes and as an iterated integral with respect to , then to , then to .

Find the area of the surface S where S is the part of the surface that lies inside the cylinder

Find the moment of inertia with respect to a diameter of the base of a solid hemisphere of radius 3 with constant mass density function

Find the mass of the solid S bounded by the paraboloid and the plane if S has constant density 3.

Set up, but do not evaluate, the iterated integral giving the mass of the solid T bounded by the cylinder in the first octant and the plane having mass density given by

Find the area of the surface S where S is the part of the sphere that lies inside the cylinder

An agricultural sprinkler distributes water in a circular pattern of radius ft. It supplies water to a depth of feet per hour at a distance of feet from the sprinkler. What is the total amount of water supplied per hour to the region inside the circle of radius feet centered at the sprinkler?

Calculate the iterated integral.

Calculate the iterated integral.

Evaluate the iterated integral by converting to polar coordinates. Round the answer to two decimal places. .