Exam 12: Multiple Integrals

Identify the surface with equation

Use the transformation to evaluate the integral ,where R is the region bounded by the ellipse .

Estimate the volume of the solid that lies above the square and below the elliptic paraboloid . Divide into four equal squares and use the Midpoint rule.

Evaluate the integral by reversing the order of integration.

Use the given transformation to evaluate the integral. ,where R is the square with vertices (0,0),(4,6),(6, ),(10,2)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 .

Use spherical coordinates. Evaluate ,where B is the ball with center the origin and radius 5.

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

Find the mass of the lamina that occupies the region D and has the given density function.Round your answer to two decimal places.

Express the triple integral as an iterated integral in six different ways using different orders of integration where T is the solid bounded by the planes and

Evaluate the double integral. is bounded by and .

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

A swimming pool is circular with a 60-ft diameter.The depth is constant along east-west lines and increases linearly from 3 ft at the south end to 9 ft at the north end.Find the volume of water in the pool.

Evaluate the iterated integral by reversing the order of integration.

Use cylindrical coordinates to evaluate

Find the moment of inertia about the y-axis for a cube of constant density 3 and side length 6 if one vertex is located at the origin and three edges lie along the coordinate axes.

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

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

Find the mass of the solid E,if E is the cube given by and the density function is .