Chapter 14.8, Problem 26E

### 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

# Finding Volume Using a Change of Variables In Exercises 23-30, use a change of variables to find the volume of the solid region lying below the surface z = f ( x , y ) and above the plane region R. f ( x , y ) = ( x + y ) 2 sin 2 ( x − y ) R: region bounded by the square with vertices ( π ,   0 ) ,   ( 3 π / 2 ,   π / 2 ) ,       ( π , π ) ,     ( π / 2 ,   π / 2 )

To determine

To calculate: The volume lying below the surface f(x,y)=(x+y)2sin(xy) and above the plane region R using change of variables.

Explanation

Given: The function f(x,y)=(x+y)2sin(xâˆ’y)

R: Region bounded by the square with coordinates,

(x,y)=(Ï€,0)(x,y)=(3Ï€2,Ï€2)(x,y)=(Ï€,Ï€)(x,y)=(Ï€2,Ï€2)

Formula used: Using Jacobian formula Î´(x,y)Î´(u,v)=|Î´xÎ´uÎ´xÎ´vÎ´yÎ´uÎ´yÎ´v|

And, change of variables for double integrals

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

We use the slope-intercept form of equation of line y=mx+c.

Where â€˜mâ€™ is the slope of the line and m=y2âˆ’y1x2âˆ’x1.

Calculation: Assuming, x=uâˆ’v2 and y=u+v2

Givesus the value of u and v as,

u=x+yâ‹¯(Î‘)v=âˆ’x+yâ‹¯(B)

Calculating the Jacobian as follows,

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

A graph is with the help of the given conditions using the slope-intercept form of equation of line y=mx+c.

Here â€˜mâ€™ is the slope of the line i.e. m=y2âˆ’y1x2âˆ’x1

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