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Chapter 13: Q. 21 (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�02鈭�04-y2ex2+y2dxdy

Short Answer

Expert verified

蟺(e4-1)4

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=rcos胃y=rsin胃x2+y2=r2dxdy=rdrd胃

The equivalent polar integral of the given integral is,

鈭�02鈭�04-y2ex2+y2dxdy鈬�鈭�0蟺/2鈭�02er2rdrd胃

03

Calculate integral

I=鈭�0蟺/2鈭�02er2rdrd胃I=鈭�0蟺/212er202d胃I=12(e4-1)鈭�0蟺/2d胃I=12(e4-1)胃0蟺/2I=蟺(e4-1)4

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