   Chapter 15.9, Problem 26E

Chapter
Section
Textbook Problem

Evaluate the integral by making an appropriate change of variables.26. ∬R sin(9x2 + 4y2) dA, where R is the region in the first quadrant hounded by the ellipse 9x2 + 4y2 = 1

To determine

To evaluate: The integral by change of variable method.

Explanation

Given:

The integral is Rsin(9x2+4y2)dA where the region in the first quadrant is enclosed by ellipse 9x2+4y2=1

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:

Rewrite the given integral as, Rsin((3x)2+(2y)2)dA

Let u=3x and v=2y

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

Find the partial derivative of x  and y with respect to u and v. u=3x then ux=3 and uy=0 and v=2y then vx=0 and vy=2.

T1=(u,v)(x,y)=|uxuyvxvy| (3)

Substitute the corresponding values in equation (3),

T1=|3002|=3(2)0=60=6

This is in the form of inverse of Jacobian transformation (T1) but obtain Jacobian transformation (T)

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