   Chapter 5.2, Problem 57E

Chapter
Section
Textbook Problem

# Find the volume of the described solid S.The base of S is the same base as in Exercise 56, but cross-sections perpendicular to the x-axis are squares.

To determine

To find:

The volume of a solid S whose base is the triangular region with vertices 0, 0, 1, 0 and 0, 1 and cross-sections perpendicular to the x-axis are squares.

Explanation

1) Concept:

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

2) Given:

Vertices of the base triangle are 0, 0, 1, 0 and 0, 1.

3) Calculations:

A cross section represents the side s(red line) of the  square.

The base of the cross section corresponding to the point  x has length 1-x. Because cross sections are perpendicular to the x-axis and they are given by the distance between lines y=0 and y=x-1.

The area of cross section that is the area of square is Ax=side2=x-12

Cross section are perpendicular to x-axis, therefore, we shall integrate cross sections with respect to x-axis

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