Chapter 14.8, Problem 17E

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347

Chapter
Section

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347
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 âˆ¬R4(x2+y2)dA

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

The following graph:

Formula used:

With the help of Jacobianâ€™s formula Î´(x,y)Î´(u,v)=|Î´xÎ´uÎ´xÎ´vÎ´yÎ´uÎ´yÎ´v|

And change of variables for double integrals is given as

âˆ¬Rf(x,y)dxdy=âˆ¬Sf(g(u,v),h(u,v))|Î´(x,y)Î´(u,v)|dudv

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

Î´(x,y)Î´(u,v)=|Î´xÎ´uÎ´xÎ´vÎ´yÎ´uÎ´yÎ´v|Î´(x,y)Î´(u,v)=|121212âˆ’12|Î´(x,y)Î´(u,v)=âˆ’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(uâˆ’v)

In the same way, substitute,

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

Thus, different values of u and are:

(u,v)=(1,âˆ’1)(u,v)=(1,1)(u,v)=(âˆ’1,1)(u,v)=(âˆ’1,âˆ’1)

The graph obtained is,

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