## Exercise ​

Evaluate the multiple integral by iterated integration, where:

1. ,
2. ,
3. , .
4. ,
5. ,

## Exercise ​

Let .

Evaluate the following integral:

Hint: integrals of the form are equal to the product of integrals.

## Exercise ​

Let be the triangular region with vertices , , . Evaluate integral

## Exercise ​

Let be the region bounded by the curves and the line segment .

Sketch and evaluate the integral

## Exercise ​

Let be the region bounded by the four curves and . Sketch and find its centroid.

## Exercise ​

Find the center of mass of trapezoidal object with corners , , and and density .

## Exercise ​

Use polar coordinates to evaluate

Hint:

## Exercise ​

In this question we will evaluate the integral

where the integral is over the half ellipsoid

We choose a change of coordinates , , .

1. Find the Jacobian of this transformation.
2. Find the set of which corresponds to .
3. Using these, evaluate the integral .

## Exercise ​

The set is a cone of height with base in the -plane.

The set is a cylinder.

Find the volume of the set which is the subset of the cone which is contained within the cylinder .

Hint: Use polar coordinates , to write the volume of as

for some , .

## Exercise ​

Let be the solid bounded above by the sphere and below by the cone .

Find the volume of .