Calculus 3 : Double Integrals

Study concepts, example questions & explanations for Calculus 3

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Example Questions

Example Question #9 : Double Integration Over General Regions

Evaluate the double integral

 

Possible Answers:

Correct answer:

Explanation:

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

Example Question #10 : Double Integration Over General Regions

Evaluate the integral 

Possible Answers:

Correct answer:

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: 

Possible Answers:

Correct answer:

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:

Possible Answers:

Correct answer:

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 

Possible Answers:

Correct answer:

Explanation:

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

Example Question #194 : Double Integrals

Evaluate the double integral

Possible Answers:

Correct answer:

Explanation:

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

Example Question #195 : Double Integrals

Evaluate the double integral

Possible Answers:

Correct answer:

Explanation:

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

Example Question #196 : Double Integrals

Integrate:

Possible Answers:

Correct answer:

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

Possible Answers:

Correct answer:

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:

Possible Answers:

Correct answer:

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

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