Exam 16: Multiple Integration

a) Change the order of integration in the integral . b) Evaluate the integral.

A lamina bounded by the curves , , and has mass density . Find the mass of the lamina.

Evaluate the double integral of the function over the rectangle. A) B)

Compute the Jacobian of the map as a function of and .

Evaluate the integral .

Let a) Sketch the region of integration and reverse the order of integration. b) Compute the integral for .

Rewrite the following integral in the order .

Let . A) Rewrite the integral in the order B) Compute the integral in the preferable order.

The value of is which of the following?

Compute where is the region .

Find if is the region satisfying and

Rewrite the integral using cylindrical coordinates and evaluate the integral.

The value of the integral is which of the following?

Find the volume of the region under the surface and above the rectangle

Let where is the tetrahedron with vertices at , , and Write as an iterated integral in the orders and .

Given the integral : a) sketch the region of integration and reverse the order of integration. b) evaluate the integral.

Calculate the probability that where and are random variables with joint probability density .

The solid in the first octant bounded by the sphere above, by the cone below, and by the planes and on the side, has a density . Set up the integral for the mass of the solid A) in rectangular coordinates. B) in cylindrical coordinates. C) in spherical coordinates.

Evaluate , where is the triangle with vertices and .

Integrate over the region in the first octant above and below