   Chapter 14.8, Problem 17E

Chapter
Section
Textbook Problem

Evaluating a Double Integral Using a Change of Variables In Exercises 17-22, use the indicated change of variables to evaluate the double integral. ∫ R ∫ 4 ( x 2 + y 2 )   d A x = 1 2 ( u + v ) y = 1 2 ( u − v ) To determine

To calculate: The value of double integral R4(x2+y2)dA using the indicated change of variables.

Explanation

Given: The provided double integral R4(x2+y2)dA

The equations, x=12(u+v) and y=12(uv).

The following graph:

Formula used:

With the help of Jacobian’s formula δ(x,y)δ(u,v)=|δxδuδxδvδyδuδyδv|

And change of variables for double integrals is given as

Rf(x,y)dxdy=Sf(g(u,v),h(u,v))|δ(x,y)δ(u,v)|dudv

Calculation: Find the value of the Jacobian as shown below.

δ(x,y)δ(u,v)=|δxδuδxδvδyδuδyδv|δ(x,y)δ(u,v)=|12121212|δ(x,y)δ(u,v)=12

Put different values of x and y coordinates from the given graph and replace them in the given 2 equations.

Substitute (x,y)=(0,1) to get,

0=12(u+v) And 1=12(uv)

In the same way, substitute,

(x,y)=(1,0)(x,y)=(0,1)(x,y)=(1,0)

Thus, different values of u and are:

(u,v)=(1,1)(u,v)=(1,1)(u,v)=(1,1)(u,v)=(1,1)

The graph obtained is,

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