# Calculus 3 : Double Integration over General Regions

## Example Questions

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### Example Question #1 : Double Integration Over General Regions

Calculate the following Integral.

Explanation:

Lets deal with the inner integral first.

Now we evaluate this expression in the outer integral.

### Example Question #2 : Double Integration Over General Regions

Calculate the definite integral of the function , given below as

Cannot be solved.

Explanation:

Because there are no nested terms containing both  and , we can rewrite the integral as

This enables us to evaluate the double integral and the product of two independent single integrals.  From the integration rules from single-variable calculus, we should arrive at the result

.

### Example Question #3 : Double Integration Over General Regions

Evaluate the following integral on the region specified:

Where R is the region defined by the conditions:

Explanation:

### Example Question #1 : Double Integration Over General Regions

Evaluate:

Explanation:

Because the x and y terms in the integrand are independent of one another, we can move them to their respective integrals:

We used the following rules for integration:

### Example Question #5 : Double Integration Over General Regions

Evaluate the following integral.

Explanation:

First, you must evaluate the integral with respect to y (because of the notation ).

Using the rules of integration, this gets us

.

Evaluated from y=2 to y=3, we get

.

Integrating this with respect to x gets us , and evaluating from x=0 to x=1, you get  .

### Example Question #671 : Multiple Integration

Compute the following integral:

Explanation:

First, you must evaluate the integral with respect to y and solving within the bounds.

In doing so, you get  and you evaluate for y from 0 to 2.

This gets you

.

This time evaluating the integral with respect to x gets you

.

Evaluating for x from 1 to 2 gets you

.

### Example Question #672 : Multiple Integration

Evaluate the double integral.

Explanation:

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

### Example Question #673 : Multiple Integration

Evaluate the double integral.

Explanation:

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

### Example Question #674 : Multiple Integration

Evaluate the double integral

Explanation:

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

### Example Question #675 : Multiple Integration

Evaluate the integral