# Calculus 3 : Double Integrals

## Example Questions

### Example Question #9 : Double Integration Over General Regions

Evaluate the double integral

Explanation:

When solving double integrals, we compute the integral on the inside first.

### Example Question #10 : Double Integration Over General Regions

Evaluate the integral

Explanation:

First, you must evaluate the integral with respect to x. This gets you  evaluated from  to . This becomes . Solving this integral with respect to y gets you . Evaluating from  to , you get .

### Example Question #191 : Double Integrals

Evaluate the following integral:

Explanation:

First, you must evaluate the integral with respect to z. Using the rules for integration, we get  evaluated from  to . The result is . This becomes , evaluated from  to . The final answer is .

### Example Question #192 : Double Integrals

Evaluate:

Explanation:

To evaluate the iterated integral, we start with the innermost integral, evaluated with respect to x:

The integral was found using the following rule:

Now, we evaluate the last remaining integral, using our answer above from the previous integral as our integrand:

The integral was found using the following rule:

### Example Question #193 : Double Integrals

Evaluate the double integral

Explanation:

To evaluate the double integral, compute the inside integral first.

### Example Question #194 : Double Integrals

Evaluate the double integral

Explanation:

aTo evaluate the double integral, compute the inside integral first.

### Example Question #195 : Double Integrals

Evaluate the double integral

Explanation:

To evaluate the double integral, compute the inside integral first.

### Example Question #196 : Double Integrals

Integrate:

Explanation:

To perform the iterated integration, we must work from inside outwards. To start we perform the following integration:

This becomes the integrand for the outermost integral:

### Example Question #197 : Double Integrals

Explanation:

To perform the iterated integral, we work from inside outwards.

The first integral we perform is

This becomes the integrand for the outermost integral,

### Example Question #198 : Double Integrals

Solve:

Explanation:

To evaluate the iterated integral, we must work from inside outward.

The first integral we evaluate is

This becomes the integrand for the outermost integral.

The final integral we evaluate is