Exam 16: Multiple Integration

Use polar coordinates to compute , where is the region in the first quadrant enclosed by the circles and .

Compute .

Consider the integral . A) Sketch the region of integration B) Interchange the order of integration. C) Evaluate the integral.

Compute where is the region .

Evaluate the double integral of the function over the rectangle

Evaluate , where is the rectangle .

Compute the integral .

Use cylindrical coordinates to compute the volume of the solid enclosed by the surfaces and .

Evaluate .

Let be the region Using polar coordinates for the integral A) find the limits of . B) find the limits of . C) convert the integral to polar coordinates.

Let be the solid occupying the region Find the moment of inertia about the axis if has density

Evaluate where is the region enclosed by the planes , and .

Define the function Find the value of which makes a probability density function.

Let be the following function: . a) Compute and . b) Is Fubini's Theorem valid in this case? If not, explain why.

Use cylindrical coordinates to compute the volume of the solid .

Let be the following function on the rectangle : . A) ? A) Compute B) Can you find in such that is equal to the value in C) Explain why there is no contradiction with the Mean Value Theorem.

Evaluate , where is the region bounded by the lines , and .

Evaluate .

The value of the integral is which of the following?

Find the volume of the solid which lies under the surface and above the rectangle .