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 October 16th, 2011, 07:17 PM #1 Senior Member   Joined: Sep 2011 From: New York, NY Posts: 333 Thanks: 0 Right Circular Cone Find the volume V of the described solid S. A frustum of a right circular cone with height h, lower base radius R, and top radius r This one I am not sure about. I guess they are just asking for a general formula because they dont give any numbers? My best guess is to integrate the volume of a cylinder from R to r: $V= \int_r^R \, 2\pi r^2 h dr$ but I think that is reaching at best.
 October 16th, 2011, 07:43 PM #2 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,155 Thanks: 466 Math Focus: Calculus/ODEs Re: Right Circular Cone Think of it as a volume by slicing, where we slice the frustum into a stack of disks: $V=\pi\int_0\,^h x^2\,dy$ x will be a linear function of y, passing through the points (R,0) and (r,h): $x^2=$$\frac{r-R}{h}y+R$$^2$ thus we have: $V=\pi\int_0\,^h $$\frac{r-R}{h}y+R$$^2\,dy$ Let $u=\frac{r-R}{h}y+R\:\therefore\:du=\frac{r-R}{h}\,dy$ giving: $V=\frac{\pi h}{r-R}\int_R\,^{r}u^2\,du=\frac{\pi h}{3(r-R)}$u^3$_R^r=\frac{\pi h}{3(r-R)}$$r^3-R^3$$=\frac{\pi h}{3}$$r^2+rR+R^2$$$
 October 16th, 2011, 07:58 PM #3 Senior Member   Joined: Sep 2011 From: New York, NY Posts: 333 Thanks: 0 Re: Right Circular Cone Impressive. The solution is so much more involved that I expected.
October 19th, 2011, 12:10 AM   #4
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Quote:
 Originally Posted by aaron-math Find the volume V of the described solid S.
Can the standard formula base area × height/3 for the volume of a cone be used? It would make this problem easy.

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# use calculus to find the volume of a frustum of a right circular cone with height , lower base radius , and top radius .

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