only HANDWRITTEN answer needed ( NOT TYPED) The volume of the solid obtained by rotating the region bounded byx = y², x = 2yabout the liney = 2can be computed using the method of washers or disks via an integralbV =- L₁ = [ (2² - 4 ) ²³ - (2-√x ) ²]with limits of integration a = 0 and b = 4The volume of this solid can also be computed using cylindrical shells via an integralV == ₁² 2n[ (v) (2y-1²)] dyαwith limits of integration α = 0 and ß:=dx v2

Given: A solid obtained by rotating the region bounded by x=y2 , x=2y about y=2. To Find : a) Using…

Answered: The volume of the solid obtained by…

Elementary Geometry For College Students, 7e

7th Edition

ISBN:9781337614085

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

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

Chapter9: Surfaces And Solids

Section9.CR: Review Exercises

Problem 22CR

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For some regions, both the washer and shell methods work well for the solid generated by revolving the region about the coordinate axes, but this is not always the case. When a region is revolved about the y-axis, for example, and washers are used, we must integrate with respect to y. It may not be possible, however, to express the integrand in terms of y. Compute the volume of the solid generated by revolving the triangular region bounded by the lines 2y = x + 4, y = x, and x = 0 about the x-axis using the washer method.
The volume of the solid obtained by rotating the region bounded by y=(x^2), y=3x about the line y=9 can be computed using cylindrical shells via an integral V=∫_______________dy (with lower limit of alpha and upper limit of beta) with limits of integration alpha=0 and beta=9 P.S. I have already asked this question but the given answer of V=∫(2pi)(y-9)(sqrt(y)-(y/3)) is incorrect.
The volume of the solid obtained by rotating the region bounded by y=x^2, y=3x about the line x = 3 can be computed using the method of washers via an integral V= ∫________________ (with lower limit of a and upper limit of b) with limits of integration a=_________ and b=_________ The volume of this solid can also be computed using cylindrical shells via an integral V= ∫_______________ (with lower limt of alpha and upper limit of beta) with limits of integration alpha=_________ and beta=__________ In either case, the volume is V=______________ cubic units
For some regions, both the washer and shell methods work well for the solid generated by revolving the region about the coordinate axes, but this is not always the case. When a region is revolved about the y-axis, for example, and washers are used, we must integrate with respect to y. It may not be possible, however, to express the integrand in terms of y. In such a case, the shell method allows us to integrate with respect to x instead. Compute the volume of the solid generated by revolving the region bounded by y = x and y = x^2 about each coordinate axis using the shell method.

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