Surface Area Problem caused dilemma in my AP Calculus BC class

Hey guys. I could some help. This problem caused an issue in my class and I just want to see how everyone goes about doing it and the answer they get:

Find the surface of area of the solid generated from rotating $\sin(x)$ on the interval of 0 to $\pi$ around the $y$ -axis.

#Calculus

Easy Math Editor

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Comments

I think that the method of cylindrical shells would work best here. The volume $V$ will be

$V = \displaystyle\int_{0}^{\pi} 2\pi xy dx = 2\pi \int_{0}^{\pi} x\sin(x) dx$ .

Evaluating by parts, we let $u = x, dv = \sin(x) dx \Longrightarrow du = dx, v = -\cos(x)$ , and so

$V = 2\pi(-x\cos(x) + \displaystyle\int \cos(x) dx) = 2\pi(-x\cos(x) + \sin(x)$ ),

which when evaluated from $x = 0$ to $x = \pi$ comes out to $V = 2\pi (-\pi \cos(\pi) - 0) = \boxed{2\pi^{2}}$ .

We could also have used the disc method but that would have been a bit more complicated, as the integral would be of the arcsin function. What was the issue within your class?

Brian Charlesworth - 5 years, 3 months ago

I'm sorry -- but we were attempting to solve for the surface area, not the volume!

Andrew Tawfeek - 5 years, 3 months ago

Oh! Sorry. I don't know why I read "surface area" as "volume" the first go around. For the part of the solid generated that is strictly above the $xz$ -plane the surface area integral is

$A = \displaystyle\int_{0}^{\pi} 2\pi x\sqrt{1 + \cos^{2}(x)} dx$ ,

which I believe can only be solved using elliptic integrals, which would be beyond the scope of an AP course. So now I understand what the issue in your class was. (WolframAlpha gives a value of 37.7038.)

Brian Charlesworth - 5 years, 3 months ago

@Brian Charlesworth – Ahh yes, thank you! There was a whole debacle about how to go about solving this and I didn't post the steps that were taken in the case that they were inaccurate. :)

Andrew Tawfeek - 5 years, 3 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$