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Given a diagram from Calculus of a Single Variable by Larson and Edward (9th edition):

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I am interested in finding the volume of various regions when rotated about various lines. Specifically, I am wondering if my set-up for finding the volume is correct; I have no issues simply integrating such functions. Note that this is not homework - I am just reviewing for an exam tomorrow.

Problem 1: $R_3$ about $x = 1$

I used a horizontal slice (disk), so my integral was $$\pi\int_0^1 (1-\sqrt{y})^2 dy$$ since every part of of the region $R_3$ is touching the axis of revolution.

Problem 2: $R_2$ about $x = 1$

Again, I used a horizontal slice, except this one was a washer. My integral was $$\pi\int_0^1 (1-y)^2 - (1-\sqrt{y})^2 dy$$

Just looking for a confirmation that this or correct or (if needed) an explanation of why I am wrong. Thanks.

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    For problem 2, check the outer radius expression in the integrand.2012-04-27
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    Ah, the outer radius is always at a distance $y$. My inner radius is still fine though, correct?2012-04-27
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    Yes.${}{}{}{}{}$2012-04-27
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    @DavidMitra Thanks for the quick responses.2012-04-27
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    [click me](http://www.google.com/search?q=thumbs+up&hl=en&client=firefox-a&hs=c6G&rls=org.mozilla:en-US:official&prmd=imvns&tbm=isch&tbo=u&source=univ&sa=X&ei=r_-ZT_GQCo-49gTgzNyPDw&ved=0CFwQsAQ&biw=1351&bih=1235&sei=uP-ZT8LjN4Kc8QTdifXuDg)2012-04-27
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    I was amused, ha.2012-04-27
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    Generally speaking, $$\int_0^1(1-\sqrt[n]x)^m=\int_0^1(1-\sqrt[m]x)^n=\frac1{C_{m+n}^n}=\frac1{C_{m+n}^m}=\frac{m!\cdot n!}{(m+n)!}$$2013-11-27

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