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

enter image description here

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. – David Mitra Apr 27 '12 at 2:04
Ah, the outer radius is always at a distance $y$. My inner radius is still fine though, correct? – Joe Apr 27 '12 at 2:07
Yes.${}{}{}{}{}$ – David Mitra Apr 27 '12 at 2:08
@DavidMitra Thanks for the quick responses. – Joe Apr 27 '12 at 2:08
click me – David Mitra Apr 27 '12 at 2:13

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