   Chapter 15.9, Problem 27E

Chapter
Section
Textbook Problem

Evaluate the integral by making an appropriate change of variables.27. ∬R ex + y dA, where R is given by the inequality |x| + |y| ≤ 1

To determine

To evaluate: The integral by change of variable method.

Explanation

Given:

The integral is Rex+ydA and the region is given by inequality |x|+|y|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:

Let u=x+y and v=x+y

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

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

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

Substitute the corresponding values in equation (3),

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)

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