Exam 15: Multiple Integrals

Sketch the region of integration associated with the integral

The joint density function for a pair of random variables and is given. Find the value of the constant .

Sketch the solid bounded by the graphs of the equations and , and then use a triple integral to find the volume of the solid.

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

Use cylindrical coordinates to find the volume of the solid that the cylinder cuts out of the sphere of radius 3 centered at the origin.

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

Evaluate the integral by reversing the order of integration.

Find the area of the surface. Round your answer to three decimal places.

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

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

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

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

Use a triple integral to find the volume of the solid bounded by and the planes and .

Find the average value of over the rectangle with vertices .

Use polar coordinates to find the volume of the sphere of radius . Round to two decimal places.

Find the mass and the center of mass of the lamina occupying the region R, where R is the region bounded by the graphs of and and having the mass density

Describe the region whose area is given by the integral.

Find the volume bounded by the cylinders and .

Find the center of mass of a lamina in the shape of an isosceles right triangle with equal sides of length if the density at any point is proportional to the square of the distance from the vertex opposite the hypotenuse. Assume the vertex opposite the hypotenuse is located at , and that the sides are along the positive axes.

Identify the surface with equation