   Chapter 15.9, Problem 19E

Chapter
Section
Textbook Problem

Use the given transformation to evaluate the integral.19. ∫∫R xy dA, where R is the region in the first quadrant bounded by the lines y = x and y = 3x and the hyperbolas xy = 1, xy = 3; x = u/v, y = v

To determine

To evaluate: The integral RxydA.

Explanation

Given:

The region R in the first quadrant is bounded by the lines y=x, y=3x and the hyperbolas xy=1, xy=3 and x=uv, y=v.

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)

Calculation:

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

Find the partial derivative of x and y with respect to u and v. x=uv then xu=1v and xv=uv2 and y=v then yu=0 and yv=1.

(x,y)(u,v)=|1vuv201|=1(1v)0(uv2)=1v+0=1v

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

xy=uvv=u

Find the boundary by using the given transformation

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