Exam 12: Multiple Integrals

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

Find the volume under and above the region bounded by and .

Use cylindrical or spherical coordinates, whichever seems more appropriate, to evaluate where E lies above the paraboloid and below the plane .

Find the mass and the center of mass of the lamina occupying the region R, where R is the triangular region with vertices and , and having the mass density

Sketch the solid whose volume is given by the iterated integral

Evaluate the double integral by first identifying it as the volume of a solid.

Find the Jacobian of the transformation.

Use polar coordinates to find the volume of the solid bounded by the paraboloid and the plane .

Evaluate the double integral. is bounded by and .

Evaluate the triple integral. Round your answer to one decimal place. lies under the plane and above the region in the -plane bounded by the curves , and .

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 .

An electric charge is spread over a rectangular region Find the total charge on R if the charge density at a point in R (measured in coulombs per square meter) is

The double integral , where , gives the volume of a solid. Describe the solid.

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

Find the mass of a solid hemisphere of radius 5 if the mass density at any point on the solid is directly proportional to its distance from the base of the solid.

A lamina occupies the part of the disk in the first quadrant. Find its center of mass if the density at any point is proportional to its distance from the x-axis.

A cylindrical drill with radius is used to bore a hole through the center of a sphere of radius . Find the volume of the ring-shaped solid that remains. Round the answer to the nearest hundredth.

Find the center of mass of a homogeneous solid bounded by the paraboloid and

Evaluate the triple integral. Round your answer to one decimal place.

Calculate the double integral. Round your answer to two decimal places.