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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
Textbook Problem

Find the volume of the described solid S.

A frustum of a pyramid with square base of b square top of side a, and height h

Chapter 5.2, Problem 50E, Find the volume of the described solid S. A frustum of a pyramid with square base of b square top of

What happens if a = b ? What happens if a = 0 ?

To determine

To find:

The volume of a frustum of a pyramid with square base of side b, square top of side a, and height h.

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 a continuous function, then the volume of S is

V=limni=1nAxi*x=abAxdx 

2) Given:

Side of Base Squareis  b, Side of Top Square is a and height is h.

3) Calculations:

From the above figure

ABC  ADE

By property of similarity of triangles

ADAB=DEBC

From the figure, substitute the values

xh=s-a2b-a2= s-ab-a

Solving for s,

s= b-ahx+a

Thus, the cross sectional area of the solid is given by

Ax=s2= b-ahx+a2

From the figure, limits of integral are from 0 to h.

So the volume of frustum is

V= 0hAxdx

V=0hb-ahx+a2dx

After expanding the square term

V= 0h[ b-ah2x2+ 2ab-2a2hx+a2] dx

After integrating

V=b-a2h2x33+ ab-a2h

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