   Chapter 5.2, Problem 62E

Chapter
Section
Textbook Problem

# The base of S is a circular disk with radius r. Parallel cross-sections perpendicular to the base are isosceles triangles with height h and unequal side in the base.(a) Set up an integral for the volume of S.(b) By interpreting the integral as an area, find the volume of S.

To determine

(a)

To find:

An integral for the volume of S

Explanation

1) Concept:

Let S be the solid that lies between x=a to 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=1nA(xi*)x=abA(x)dx

2) Calculation:

Area of triangle of height h is

A=12·b·h

where, b is base and h is height

Equation of circular disk, centered at origin of radius r is  x2+y2=r2

Now given that the triangle is perpendicular to the base of a circular disk, so the base of the triangle is at y is

AB=2y

But,  y=r<

To determine

(b)

To find:

The volume of S

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