   Chapter 5.R, Problem 24E

Chapter
Section
Textbook Problem

# The base of a solid is the region bounded by the parabolas y = x 2 and y = 2 − x 2 . Find the volume of the solid if the cross-sections perpendicular to the x -axis are squares with one side lying along the base.

To determine

To find:

The volume of solid S whose base is bounded by the parabolas y=x2 and y=2-x2 and cross sections perpendicular to the x-axis are squares with one side lying along the base.

Explanation

1) Concept:

Use the definition of volume.

2) Definition of volume:

Let S be a solid that lies between x=a and  x=b. If the cross sectional area of S in the plane Px, through x and perpendicular to the x-axis, is  A(x), where A is continuous function, then the volume of S is

V=limni=1nAxi*x=abAxdx

3) Calculations:

The base of S is bounded by the parabolas y=x2 and y=2-x2 as shown in the above figure.

And cross sections perpendicular to the x-axis are squares with one side lying along the base (black line).

The solid is made up of squares with side lengths that are vertical distances between the two curves.

From the figure, it can be seen that it is symmetrical. So integrate from 0 to 1, and double it.

Vertical distance between two curves =2-x2-x2=21-x2

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