   Chapter 15.9, Problem 23E

Chapter
Section
Textbook Problem

Evaluate the integral by making an appropriate change of variables.23. ∬ R x − 2 y 3 x − y dA. Where R is the parallelogram enclosed by the lines x – 2y = 0, x – 2y = 4, 3x – y = 1, and 3x – y = 8

To determine

To evaluate: The integral by change of variable method.

Explanation

Given:

The integral is Rx2y3xydA and the parallelogram is enclosed by the lines x2y=0, x2y=4, 3xy=1 and 3xy=8.

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)

Definition used: Jacobian transformation

(x,y)(u,v)=|xuxvyuyv|=xuyvxvyu

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:

Let u=x2y and v=3xy.

Obtain the value of |(x,y)(u,v)|.

Find the partial derivative of x  and y with respect to u and v. u=x2y then ux=1 and uy=2 and v=3xy then vx=3 and vy=1

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