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Radmetalmonk
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Homework Statement
Find the volume generated by rotating the given region about the specified line
The region, R2, is y = x^1/2 (y = sqrt(x) )
The line, AB, is x = 1
So it's y = x^1/2 rotated about x = 1
I've solved this equation easily numerous times using cylindrical shells and just cylinders with respect to dy and I get 8pi/15 every time, while the book claims it's 7pi/15
Homework Equations
Cylindrical shells method:
V(x) = 2pi r h dr so V(x) = 2pi(1 - x)(x^1/2)dx
Or, with cylinders...
If y = x^1/2 then x = y^2
So if V(x) = pi r^2 dr then
V(x) = pi (1-x)^2 dx Substitute y for x you get V(y) = pi {1-(y^2)}^2 dy
The Attempt at a Solution
Cylindrical shell style...(integral where b = 1, a = 0)
V(x) = 2pi(1-x)(x^1/2) dx
2pi |x^1/2 - x^3/2 dx
(x^3/2)/(3/2) - (x^5/2)/(5/2) = (2x^3/2)/3 - (2x^5/2)/5
(2(1)^3/2)/3 - (2(1)^5/2)/5 = 2/3 - 2/5 = 10/15 - 6/15 = (4/15)*2pi = 8pi/15 which is incorrect, the book says it's 7pi/15
Disk method...(integral where b = 1, a = 0)
V(y) = pi {1-(y^2)}^2 dy
(1-y^2)(1-y^2) = 1 - y^2 - y^2 + y^4 = 1 - 2y^2 + y^4 Integral it...
pi| y - 2y^3/3 + y^5/5 = 1 - (2(1)^3)/3 + ((1)^5)/5 = 15/15 - 10/15 + 3/15 = 8/15*pi = 8pi/15 which is incorrect, the book says it's 7pi/15
Sorry for the messy work, any help would be appreciated! Maybe it's not the integral from 0 to 1? Perhaps 2 to 1? I'm going to try that next. Also the book may be wrong, I certainly hope that if that's the case that the grader knows that as well!
EDIT: Alright, I'm pretty sure the book is wrong on this one. I've been doing a group of problems concerning the same 3 regions divided by the same 2 functions rotated about the same lines. One of them is the region between R1 and R2. I found R1 (it's the region below y = x^3, so when rotated about AB (x =1 ) it's volume was pi/10 and the book agrees with that. R2, my problem that I posted here, when used as what the book says it is, 7pi/15, I get the volume between these regions as 11pi/30 (work: 70pi/150 - 15pi/150 = 55pi/150 = 11pi/30) while the book says it's 13pi/30. Well guess what? If I use my answer 8pi/15 instead of 7pi/15, I get the correct answer. So I'm pretty sure that proves that the book screwed up in this case. But just to be sure, tell me what you guys get. Thanks!
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