   Chapter 15.9, Problem 25E

Chapter
Section
Textbook Problem

Evaluate the integral by making an appropriate change of variables.25. ∬ R cos ( ) dA, where R is the trapezoidal region with vertices (1, 0), (2, 0), (0, 2), and (0, 1)

To determine

To evaluate: The integral by change of variable method.

Explanation

Given:

The integral is Rcos(yxy+x)dA where the trapezoidal region with vertices (1,0),(2,0),(0,2) and (0,1).

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

Calculation:

From the given integral observe and let u=yx and v=y+x.

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

Find the partial derivative of x  and y with respect to u and v. u=yx then ux=1 and uy=1 and v=y+x then vx=1 and vy=1.

T1=(u,v)(x,y)=|uxuyvxvy| (2)

Substitute the corresponding values in equation (2),

T1=|1111|=1(1)1(1)=1+1=2

This is in the form of inverse of Jacobian transformation (T1) but obtain Jacobian transformation (T). Therefore the value of Jacobian 2 will be 12

Substitute the given vertices of trapezoidal in u=yx and v=y+x to get the image of the trapezoidal with vertices that is (1,1),(2,2),(2,2) and (1,1)

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