# Determining Volume by rotation of a plane

a) What is the area of the region R between the graphs of $y= \sin x$ and $y=\sin ^2 x$ for $x \in [0; \frac{\pi}{2}]$

I found $a(R)= 1-\frac{\pi}{4}$

b) Let R be the region from question a). Rotating $R$ about the x-axis describes a solid S. Compute the volume of S

Thank you

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Whoops, you're right... Why ask then? –  Karolis Juodelė Apr 14 '13 at 16:14
Yes, both I and Xiaolang made a mistake. The area of a circle of radius $r_1$ with a hole of radius $r_2$ is not the same as the area of a circle of radius $r_1 - r_2$. The idea is the same whether it's $\sin x^2$ or $\sin^2 x$. In the latter case the integration will be easier though. –  Karolis Juodelė Apr 14 '13 at 16:22
@KarolisJuodelė Ok thank you. Could you help me with a similar question here regarding the concept of a "mass" math.stackexchange.com/questions/361342/mass-of-a-rectangle/… –  user43418 Apr 14 '13 at 16:25
yes. (the comment area isn't really for this. delete your comments after you've solved the problem and flag mine). –  Karolis Juodelė Apr 14 '13 at 16:25
–  user43418 Apr 14 '13 at 16:25

just like the Riemann integrate definition you can put the area of the section as a contionous function $f(x)$

and then calculate the integration $\int_a^bf(x)dx$ it is just the volume

and in advance to calculate a rotating body's volume

the $f(x)$ can be written by the function in (a) as $f(x)=\pi g(x)^2$ ($g(x)$ acts as a "radius")

and the sums changes to $V=\pi \int_a^bg^2(x)dx$

in the OP's question ,the $g(x)$ is equals to $\sin x-\sin x^2$

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But what is the exact expression of the integral. And what are the limits of the integral ? –  user43418 Apr 14 '13 at 14:52
@user43418 sorry i put the "send" button in accident and edit it later... –  Xiaolang Apr 14 '13 at 14:54
the $f(x)$ refer to the area of the section,the first formula is more common and fundamantal and the second formula especially used to calculate the rotating body's volume –  Xiaolang Apr 14 '13 at 14:57
@Xiaolang What is g ? –  Jean-Francois Rossignol Apr 14 '13 at 14:57
@Jean-FrancoisRossignol g(x) is like a radius in the circle. –  Xiaolang Apr 14 '13 at 14:58