Chapter 14.8, Problem 29E

### 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 R: region bounded by the square with vertices (0, 0), (a, 0), (0, a), where a > 0

To determine

To calculate: The volume lying below the surface f(x,y)=x+y and above the surface Rusing change of variables.

Explanation

Given: The function f(x,y)=x+y

R: Region bounded by the triangle with vertices,

(x,y)=(0,0)(x,y)=(a,0)(x,y)=(0,a)

Formula used: Applying 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=u+v2 and y=uâˆ’v2

It givesus the value of u and v as,

u=x+yâ‹¯(Î‘)v=xâˆ’yâ‹¯(B)

Calculatin the Jacobian as,

Î´(x,y)Î´(u,v)=|Î´xÎ´uÎ´xÎ´vÎ´yÎ´uÎ´yÎ´v|Î´(x,y)Î´(u,v)=|12âˆ’123414|Î´(x,y)Î´(u,v)=âˆ’12

With the help of the given conditions, a graph is constructed using the slope-intercept form of equation of line y=mx+c

### Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

#### The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started