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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.

-50-25-x2034+x2+y23dydx

Short Answer

Expert verified

-50-25-x2034+x2+y23dydx=2475107648π

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 following substitution,

x=rcosθy=rsinθx2+y2=r2dxdy=rdrdθ

The equivalent polar integral of the given integral is,

-50-25-x2034+x2+y23dydxπ3π20-53(4+r2)3rdrdθ

03

Calculate the volume

">V=π3π20-53(4+r2)3rdrdθV=π3π2dθ0-53(4+r2)3rdrV=3π2-π0-53(4+r2)3rdrV=π20-53(4+r2)3rdrSubstitute,4+r2=t2rdr=dtdr=12rdtWhenr=0,t=4Whenr=-5,t=4+(-5)2=29 V=π2324291t3dtV=3π4t-3+1-3+1429V=-3π81t2429V=-3π81841-116V=-3π816-84113456V=-3π8-82513456V=2475107648πcubicunits

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