   Chapter 15.2, Problem 23E

Chapter
Section
Textbook Problem

Find the volume of the given solid.23. Under the plane 3x + 2y – z = 0 and above the region enclosed by the parabolas y = x2 and x = y2

To determine

To find: The volume of the solid that lies under the plane and above the region enclosed by parabolas.

Explanation

Given:

The plane is 3x+2yz=0 .

The parabolas are, y=x2,x=y2 .

Formula used:

The volume of the solid, V=DzdA , where, z is the given function.

Calculation:

Express the given plane equation as follows:

z=3x2yz=3x+2y

The parabola x=y2 can be rewritten as y=x .

Solve both the equation of the parabola and obtain the limit for x as 0 to 1.

So, the volume of the solid is computed as follows.

V=RzdA=01x2x(3x+2y)dydx

First, compute the integral with respect to y.

V=01[3xy+2y22]x2xdx

Apply the limit value for y,

V=01[(3xx+(x)2)(

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