   Chapter 15.9, Problem 18E

Chapter
Section
Textbook Problem

Use the given transformation to evaluate the integral.18. ∫∫R (x2 – xy + y2) dA, where R is the region bounded by the ellipse x2 - xy + y2 = 2; x = 2 u - 2 / 3 v, y = 2 u + 2 / 3 v

To determine

To evaluate: The integral R(x2xy+y2)dA.

Explanation

Given:

The region R is bounded by the ellipse x2xy+y2=2 and x=2u23v,y=2u+23v

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|

Find the partial derivative of x and y with respect to u and v. x=2u23v then xu=2 and xv=23 and y=2u+23v then yu=2 and yv=23.

(x,y)(u,v)=|223223|=2(23)2(23)=23+23=43

From the given integral the function is, x2xy+y2 and substitute the values of x and y.

x2xy+y2=(2u23v)2(2u23v)(2u+23v)+(2u+23v)2=(2u)2+(23v)222u23v((2u)2(23v)2)+(2u)2+(23v)2+22u23v

Simplify the above as possible

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