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Chapter 13: Q 14. (page 1014)

Explain why the double integral 鈭�惟dAgives the area of the region惟 . Illustrate your explanation with an example.

Short Answer

Expert verified

It is solved by solving type I integral.

Step by step solution

01

Given Information

We are given double integral 鈭�惟dAover region 惟

02

Simplification

Consider arbitrary type I region bounded on left by x=a, to right by x=b

below by y=g1(x)and above by y=g2(x)

Integral 鈭�惟dAover type I is calculated by taking elementary area of width 鈭�xwith one end lying over y=g1(x)and other over y=g2(x)

The integral becomes

鈭�惟dA=鈭�abg1(x)鈭�g2(x)dydx

鈭�惟dA=鈭�ab[y]g1(x)g2(x)dx

=鈭�abg2(x)-g1(x)dx

Hence, integral鈭�惟dAgives area over region惟

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