   Chapter 14.8, Problem 29E

Chapter
Section
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 δ(x,y)δ(u,v)=|δxδuδxδvδyδuδyδv|

And, change of variables for double integrals

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

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

Where ‘m’ is the slope of the line and m=y2y1x2x1.

Calculation: Assuming, x=u+v2 and y=uv2

It givesus the value of u and v as,

u=x+y(Α)v=xy(B)

Calculatin the Jacobian as,

δ(x,y)δ(u,v)=|δxδuδxδvδyδuδyδv|δ(x,y)δ(u,v)=|12123414|δ(x,y)δ(u,v)=12

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

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