   Chapter 5.2, Problem 56E

Chapter
Section
Textbook Problem

# Find the volume of the described solid S.The base of S is the triangular region with vertices (0, 0), (1, 0), and (0, 1). Cross- sections perpendicular to the y-axis are equilateral triangles.

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 y-axis are equilateral triangles.

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 base triangle is 0, 0, 1, 0 and 0, 1.

3) Calculation:

In the figure the base of cross section at y is represented by s(red line).

Then the height of the equilateral triangle (cross section) is

h=s ·sin60=32s

So the area of the triangle is

A=12·side·height=12s·32s=34s2

The equation of the line representing the diagonal is x+y=1

That is x=-y+1

Therefore, at y  the base of cross section, s is a line joining (0,y) and (-y+1,y). Thus the value of s at y is

s(y)= -y +1

The cross section are perpendicular to y-axis

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