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I was attempting to solve this problem regarding a volume of a solid of revolution (reference: Apostol, Volume 1, Section 6.26, Question 18 (c)). The question is (paraphrased) as follows:

Let $f(x) = e^{-2x} \forall x$. Denote by $S(t)$ the ordinate set of $f$ over the interval $[0,t]$, where $t>0$. Let $W(t)$ be the volume of the solid obtained by rotating $S(t)$ about the $y$ axis. Compute $W(t)$.

My attempt at this is as follows:

$y = e^{-2x} \Rightarrow \log y = -2x \Rightarrow x = \frac{\log y}{-2}$

Which means $$ W(t) = \pi \int_{e^{-2t}}^1 \left(\frac{\log y}{-2}\right)^2 dy $$ $$ = \frac{\pi}{4}\int_{e^{-2t}}^1 \left(\log y \right)^2 dy $$ $$ = \frac{\pi}{4} \left[y(\log y)^2 - 2y(\log y -1)\right]_{e^{-2t}}^1 $$ $$ = \frac{\pi}{4} \left( (1(\log 1)^2 -2\cdot 1(\log 1 -1)) - (e^{-2t}(\log e^{-2t})^2-2e^{-2t}(\log e^{-2t}-1))\right) $$ $$ = \frac{\pi}{4} \left(2 - (4t^2e^{-2t}+4te^{-2t}+2e^{-2t})\right) $$ $$ = \frac{\pi}{4} \left(2 - 2e^{-2t}(2t^2+2t+1)\right) $$ $$ = \frac{\pi}{2}\left(1-e^{-2t}(2t^2+2t+1)\right) $$ which is where I have my issue since the answer in the back of the book is $\frac{\pi}{2}\left(1-e^{-2t}(2t+1)\right)$, so either the book is wrong or I am wrong. If anyone can shed some light on this that would be appreciated

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I would say assignment deceiving. S(t) = the arc function f(t), $t\in \langle 0,1 \rangle$.

Then the volume of the solid obtained by rotating S(t) about the y axis:

$W(t)=2\pi\int_{0}^{t}xf(x)\,dx=2\pi\int_{0}^{t}xe^{-2x}\,dx=\cdots =\frac{\pi}{2}\left(1-e^{-2t}(2t+1)\right)$.

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  • $\begingroup$ Interesting: the best reference I have found for this has been in Wikipedia (en.wikipedia.org/wiki/Solid_of_revolution) when talking about the Disc method vs. the Shell method for computing volumes. While Apostol talks about the solid of revolution using the Disc method (Section 2.12) for revolving the function about the x-axis, it seems as if he is drawing a fairly long bow when inferring about the solid of revolution when revolving around the y-axis. $\endgroup$
    – emjay
    Sep 18 '14 at 3:10
  • $\begingroup$ Sorry about my last comment: I was referring to how Apostol only seems to loosely infer about the shell method for solids of revolution. Thank you, @georg $\endgroup$
    – emjay
    Sep 18 '14 at 3:38

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