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Chapter 13: Q. 23 (page 1082)

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

鈭�-33鈭�-9-x29-x2x+2yx2+y2dydx

Short Answer

Expert verified

鈭�-33鈭�-9-x29-x2x+2yx2+y2dydx=0

Step by step solution

01

Step !: Draw the region

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

02

Convert into polar form

By using the below substitution,

x=rcos胃y=rsin胃x2+y2=r2dxdy=rdrd胃

The equivalent polar integral of the given integral is,

鈭�-33鈭�-9-x29-x2x+2yx2+y2dydx鈬�鈭�02蟺鈭�03(cos胃+2sin胃)drd胃

03

Calculate the integral

I=鈭�02蟺鈭�03(cos胃+2sin胃)drd胃I=鈭�02蟺(cos胃+2sin胃)d胃鈭�03rdrI=sin胃-2cos胃02蟺鈭�03rdrI=-2-(-2)鈭�03rdrI=(-2+2)鈭�03rdrI=0

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