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Using polar coordinates to evaluate iterated integrals: Evaluate the given iterated integrals by converting them to polar coordinates. Include a sketch of the region.

0404y-y21x2+y2dxdy

Short Answer

Expert verified

0404y-y21x2+y2dxdy=π

Step by step solution

01

Draw the region

From the limits of integration, the region is shown below,

02

Convert into polar form

By using the below substitution,

x=rcosθy=rsinθx2+y2=r2dxdy=rdrdθ

The equivalent polar integral of the given integral is,

0404y-y21x2+y2dxdy0π/224drdθ

03

Calculate integral

I=0π/224drdθI=r240π/2dθI=2θ0π/2I=2×π2I=π

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