Book edition 2nd

Author(s) James Stewart

Pages 830 pages

ISBN 9781133112280

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Short Answer

To calculate the volume of the described solid (circular disk).

The volume of the circular disk is \(\frac{{16}}{3}{r^3}\).

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Given data

The circular disk with radius \(r\) and base \(S\).

Parallel cross-sections perpendicular to the base are squares.

Concept used of Volume

Volume is a scalar quantity expressed by the amount of three-dimensional space enclosed by a closed surface.

Solve to find the volume

Consider that the Equation of circle \({x^2} + {y^2} = {r^2}\) ….(1)

Rearrange Equation (1).

\(\begin{aligned}{l}{x^2} = {r^2} - {y^2}\\y = \sqrt {{r^2} - {x^2}} \end{aligned}\)

The cross-section of the circular disk as shown in Figure 1.

The region lies between \(a = - r\) and \(b = r\).

The expression to find the volume of the circular disk as shown below.

\(V = \int_a^b A (x)dx\) …..(2)

Substitute \( - r\) for a, r for \(b\), and \(4\left( {{r^2} - {x^2}} \right)\) for \(A(x)\) in Equation (2).

\(\begin{aligned}{}V = \int_{ - r}^r 4 \left( {{r^2} - {x^2}} \right)dx\\ = 8\int_0^r {\left( {{r^2} - {x^2}} \right)} dx\\ = 8\left( {{r^2}x - \frac{{{r^3}}}{3}} \right)_0^r\\ = 8\left( {{r^2} \times r - \frac{{{r^3}}}{3} - 0} \right)\end{aligned}\)

Simplify further,

\(\begin{aligned}{}V = \frac{8}{3}\left( {3{r^3} - {r^3}} \right)\\ = \frac{{16}}{3}{r^3}\end{aligned}\)

Therefore, the volume of the circular disk is \(\frac{{16}}{3}{r^3}\).

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Essential Calculus: Early Transcendentals

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Short Answer

To calculate the volume of the described solid (circular disk).

Step by Step Solution

Given data

Concept used of Volume

Solve to find the volume

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Essential Calculus: Early Transcendentals

Answers without the blur.

Short Answer

To calculate the volume of the described solid (circular disk).

Step by Step Solution

Given data

Concept used of Volume

Solve to find the volume

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