Exam 16: Multiple Integration

Find the coordinates of the centroid for the region described by

Let be continuous. If the equality holds, then:

Compute the volume of the region satisfying and

Find the coordinates of the centroid for the sector of the unit disk satisfying where

Evaluate where is the tetrahedron with vertices at , , and in the order .

Compute , where is the parallelogram with edges , and .

The volume of the tetrahedron with the vertices and is which of the following?

Let be the solid bounded by the cylinder and the planes , and in the first octant. The density of the solid is . A) Find the mass of the solid. B) Find the center of mass of the solid.

Use cylindrical coordinates to compute the volume of the solid bounded by the paraboloid and the upper sphere .

If the mass density is find for the solid occupying the region which satisfies and

The volume of the region enclosed by the cylinder , the cone and the plane is which of the following?

Let and The value of is which of the following?

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

Evaluate the iterated integral .

The of the solid region that is inside the cylinder , below the paraboloid and above the plane (assuming ) is which of the following?

Find the volume of the region satisfying and

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

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

Find the mass and of the solid in the first octant bounded by the cylinder and the planes and , assuming that the mass density is .

Find the volume of the portion of the ball satisfying and lying above the surface defined by