Exam 15: Multiple Integrals

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

Calculate the iterated integral.

Find the center of mass of the system comprising masses m_k located at the points P_k in a coordinate plane. Assume that mass is measured in grams and distance is measured in centimeters. m₁ = 4, m₂ = 3, m₃ = 2 P₁(-3, -3), P₂(0, 3), P₃(-2, -1)

Use a double integral to find the area of the region R where R is bounded by the circle

Find the Jacobian of the transformation.

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 .

Evaluate the integral by changing to polar coordinates. is the region bounded by the semicircle and the -axis.

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

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

Find the area of the surface. The part of the surface that lies within the cylinder .

Evaluate the double integral. is bounded by and .

Use spherical coordinates to find the moment of inertia of the solid homogeneous hemisphere of radius and density 1 about a diameter of its base.

A swimming pool is circular with a -ft diameter. The depth is constant along east-west lines and increases linearly from ft at the south end to ft at the north end. Find the volume of water in the pool.

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

Evaluate the double integral , where

Evaluate the double integral , where is the triangular region with vertices and .

Evaluate the integral by changing to polar coordinates.

Evaluate the integral by reversing the order of integration.

Find the mass of the lamina that occupies the region D and has the given density function, if D is bounded by the parabola and the line .

Calculate the double integral.