Answered: find the volume of the solid generated…

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

find the volume of the solid generated by
revolving the region about the given line. The region in the first quadrant bounded above by the line y = 2,
below by the curve y = 2 sin x, 0 ≤ x ≤ π/2, and on the left by
the y-axis, about the line y = 2

Expert Solution

Step 1

Given region is

$y = 2 y = 2 \sin x A b o u t t h e l i n e y = 2$

Step 2

So required volume:

$\begin{array}{rcl} V & = & \int_{a}^{b} πA (x) d x \\ = & \int_{0}^{\frac{π}{2}} π {(2 - 2 \sin x)}^{2} d x [A (x) = {(2 - 2 \sin x)}^{2}] \\ = & \int_{0}^{\frac{π}{2}} π (4 - 8 \sin x + 4 \sin^{2} x) d x \\ = & \int_{0}^{\frac{π}{2}} π (4 - 8 \sin x + 4 (\frac{1 - \cos 2 x}{2})) d x \\ = & π \int_{0}^{\frac{π}{2}} (4 - 8 \sin x + 2 - 2 \cos 2 x) d x \\ = & π \int_{0}^{\frac{π}{2}} (6 - 8 \sin x - 2 \cos 2 x) d x \\ = & π {(6 x - 8 (- cosx) - 2 (\frac{\sin 2 x}{2}))}_{0}^{\frac{π}{2}} \\ = & π {(6 x + 8 cosx - 2 (\frac{\sin 2 x}{2}))}_{0}^{\frac{π}{2}} \\ = & π (\frac{6 π}{2} + 0 - 0 - (0 + 8 - 0)) \\ = & π (3 π - 8) \end{array}$

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