Answered: 1. Give an integral (BUT DO NOT…

1. Give an integral (BUT DO NOT INTEGRATE) to find the volume of the solid that is formed by rotating the region bounded by y = x, y = 1, and x 0 about the line x 3.

Elementary Geometry For College Students, 7e

7th Edition

ISBN:9781337614085

Author:Alexander, Daniel C.; Koeberlein, Geralyn M.

Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.

Chapter10: Analytic Geometry

Section10.1: The Rectangular Coordinate System

Problem 40E: Find the exact volume of the solid that results when the region bounded in quadrant I by the axes...

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Question

1. Give an integral (BUT DO NOT INTEGRATE) to find the volume of the solid that is formed
by rotating the region bounded by y = x, y = 1, and x
0 about the line x
3.

Expert Solution

Step 1

Concept Used

The shell method: The volume of the solid obtain by rotating the region is given by

$v o l u m e = \int 2 π (s h e l l r a d i u s) (s h e l l h e i g h t)$

Step 2

Calculation:

Given that the solid is formed by rotating region bounded by $y = x, y = 1, x = 0 a b o u t t h e l i n e x = 3$

therefore,

$\begin{array}{rcl} v o l u m e & = & \int 2 π (s h e l l r a d i u s) (s h e l l h e i g h t) \\ = & 2 π \int_{0}^{1} x (3 - x) d x \\ = & 2 π \int_{0}^{1} (3 x - x^{2}) d x \\ = & 2 π [\int_{0}^{1} 3 x d x - \int_{0}^{1} x^{2} d x] \\ = & 2 π [3 {[\frac{x^{2}}{2}]}_{0}^{1} - {[\frac{x^{3}}{3}]}_{0}^{1}] \\ = & 2 π [\frac{3}{2} - \frac{1}{3}] \\ = & 2 π \times \frac{7}{6} \\ volume & = & \frac{7 π}{3} \end{array}$

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