Calculus 3 : Double Integration in Polar Coordinates

Study concepts, example questions & explanations for Calculus 3

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Example Questions

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Example Question #1 : Double Integration In Polar Coordinates

Evaluate the following integral by converting into Polar Coordinates.

, where  is the portion between the circles of radius  and  and lies in first quadrant. 

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Explanation:

We have to remember how to convert cartesian coordinates into polar coordinates.

Lets write the ranges of our variables  and .

 

Now lets setup our double integral, don't forgot the extra .

 

 

Example Question #1 : Double Integration In Polar Coordinates

Evaluate the integral

where D is the region above the x-axis and within a circle centered at the origin of radius 2.

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Correct answer:

Explanation:

The conversions for Cartesian into polar coordinates is:

The condition that the region is above the x-axis says:

And the condition that the region is within a circle of radius two says:

With these conditions and conversions, the integral becomes:

Example Question #2 : Double Integration In Polar Coordinates

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Example Question #3 : Double Integration In Polar Coordinates

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Example Question #4 : Double Integration In Polar Coordinates

 

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Example Question #6 : Double Integration In Polar Coordinates

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Example Question #5 : Double Integration In Polar Coordinates

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Example Question #6 : Double Integration In Polar Coordinates

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Example Question #7 : Double Integration In Polar Coordinates

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Example Question #8 : Double Integration In Polar Coordinates

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