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Chapter 13: Q 62. (page 1014)

Evaluate the double integral over the specified region.

鈭�惟dA,where 惟is region in first quadrant bounded by curves y=xmandy=xnwherem,nare distinct positive integers.

Short Answer

Expert verified

The required integral is|m-n|(m+1)(n+1).

Step by step solution

01

Given Information

We are given double integral as鈭�惟dA, region is bounded by curvesy=xmandx=yn

02

Considering type I integral

The curves will intersect at x=1

For type I integral, 0鈮�x鈮�1,xm鈮�y鈮�xn

03

Simplification

The integral is written as:

鈭�惟dA=鈭�01鈭�x''xndydx

鈭�01鈭�xnxndydx=鈭�01[y]xmxndx

鈭�01鈭�xmxndydx=xn+1n+1-xm+1m+101

=1n+1-1m+1

=m-n(m+1)(n+1)

Integral is|m-n|(m+1)(n+1)

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Most popular questions from this chapter

Evaluate each of the double integral in the exercise 37-54 as iterated integrals

鈭�鈭�R(x-ey)dAWhereR={(x,y)|-3鈮�x鈮�2,-2鈮�y鈮�2}

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鈭�鈭�24鈥�鈭�26鈥�鈭�05鈥�f(x,y,z)dydxdz

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鈭�惟鈥�z蚁(x,y,z)dV

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