Calculus 3 : Double Integration over General Regions

Study concepts, example questions & explanations for Calculus 3

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Example Questions

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Example Question #1 : Double Integration Over General Regions

Calculate the following Integral.

Possible Answers:

Correct answer:

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 

 

Possible Answers:

Cannot be solved.

Correct answer:

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:

Possible Answers:

Correct answer:

Explanation:

Example Question #3 : Double Integration Over General Regions

Evaluate:

Possible Answers:

Correct answer:

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 #4 : Double Integration Over General Regions

Evaluate the following integral. 

Possible Answers:

Correct answer:

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 #5 : Double Integration Over General Regions

Compute the following integral: 

Possible Answers:

Correct answer:

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 #6 : 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 #7 : 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 #8 : 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 #9 : 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 .

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