   Chapter 15.9, Problem 20E

Chapter
Section
Textbook Problem

Use the given transformation to evaluate the integral.20. ∫∫R y2 dA, where R is the region bounded by the curves xy = 1, xy = 2, xy2 = 2; u = xy, v = xy2. Illustrate by using a graphing calculator or computer to draw R.

To determine

To evaluate: The integral Ry2dA .

Explanation

Given:

The region R bounded by the curves xy=1 , xy=2 , xy2=1 , xy2=2 and u=xy , v=xy2 .

Property used: Change of Variable

Change of Variable in double integral is given by,

Rf(x,y)dA=Sf(x(u,v),y(u,v))|(x,y)(u,v)|dudv (1)

Formula used:

If g(x) is the function of x and h(y) is the function of y then,

ababg(x)g(y)dydx=abg(x)dxabh(y)dy (2)

Calculation:

Obtain the Jacobian, (x,y)(u,v)=|xuxvyuyv|

Write u and v in terms of x and y respectively.

x=u2v and y=vu .

Find the partial derivative of x and y with respect to u and v. x=u2v then xu=2uv and xv=u2v2 and y=vu then yu=vu2 and yv=1u .

(x,y)(u,v)=|2uvvu2u2v21u|=2uv(1u)(vu2)(u2v2)=2v1v=1v

From the given integral the function is, y2 and substitute the values of x and y

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