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Let g1(x)andg2(x)be two continuous functions such that g1(x)g2(x)on the interval [a,b], and let be the region in the xy-plane bounded by g1andg2on[a,b]. Use your answer to Exercise 14 to set up an iterated integral whose value is the area of Ω. How is this iterated integral related

to the definite integral you would have used to compute the area of Ωin Chapter 4?

Short Answer

Expert verified

The iterated integral that gives area of region isabgg1(x)2g2(x)dydx

Step by step solution

01

Given Information

Two continuous functions g1(x)andg2(x)such that g1(x)<g2(x)in a,b.

Region Ωis bounded by curves g1andg2in a,b.

02

Consideration

The region bounded on left by x=aand right by x=a, y=g1(x)below and y=g1(x)above is of type I

Hence, surface integral becomes

ΩdA=abgg1(x)2g2(x)dydx

Integral that gives area of region is abg1(x)g2(x)dydx.

ΩdA=ab[y]g1(x)g2(x)dx

=abg2(x)-g1(x)dx

Hence, integralΩdAgives area of regionΩ

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