My Math Forum Volumes By Cylindrical Shells

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 October 13th, 2011, 07:21 PM #1 Senior Member   Joined: Sep 2011 From: New York, NY Posts: 333 Thanks: 0 Volumes By Cylindrical Shells
 October 13th, 2011, 08:34 PM #2 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,193 Thanks: 504 Math Focus: Calculus/ODEs Re: Volumes By Cylindrical Shells (a) Well, let's see: $R_1=\int_0\,^b x^2\,dx=\frac{1}{3}$x^3$_0^b=\frac{b^3}{3}$ $R_2=\int_0\,^b b^2-x^2\,dx=$b^2x-\frac{1}{3}x^3$_0^b=\frac{2b^3}{3}$ Equating, we find: $\frac{b^3}{3}=\frac{2b^3}{3}$ $b^3=2b^3$  Since b is given to be positive, there is no real value of b satisfying the given requirement. For all positive b, we have: $R_2=2R_1$ (b) Computing the solids of revolution: $V_1=\pi\int_0\,^b $$x^2$$^2\,dx=\pi\int_0\,^b x^4\,dx=\frac{\pi}{5}$x^5$_0^b=\frac{\pi b^5}{5}$ $V_2=\pi\int_0\,^b $$b^2$$^2-$$x^2$$^2\,dx=\pi\int_0\,^b b^4-x^4\,dx=\pi$b^4x-\frac{x^5}{5}$_0^b=\frac{4\pi b^5}{5}$ Again, there is no positive real value for b that equates the two solids of revolution. For all positive values of b we have: $V_2=4V_1$ (c) Computing the solids of revolution: $V_1=\pi\int_0\,^{b^2} (b)^2-(x)^2\,dy=\pi\int_0\,^{b^2} b^2-y\,dy=\pi$b^2y-\frac{1}{2}y^2$_0^{b^2}=\frac{\pi b^4}{2}$ $V_2=\pi\int_0\,^{b^2} (x)^2\,dy=\pi\int_0\,^{b^2} y\,dy=\frac{\pi}{2}$y^2$_0^{b^2}=\frac{\pi b^4}{2}$ For all values of b, we have: $V_1=V_2$ I used the disk/washer methods for the solids of revolution, I recommend you using the shell method for parts (b) and (c) and see if our results agree. For part (a), I recommend integrating along the y-axis for an alternate method.
 October 14th, 2011, 06:17 AM #3 Member   Joined: Oct 2011 Posts: 45 Thanks: 0 Re: Volumes By Cylindrical Shells can it be done by changing the number sequence
 October 14th, 2011, 03:56 PM #4 Senior Member   Joined: Sep 2011 From: New York, NY Posts: 333 Thanks: 0 Re: Volumes By Cylindrical Shells Lets see what I can do: a) Integrating with respect to $y$ I think the new bounds for $R_1$would be from zero to $f(b)$, and I think this would give me the value of the area shaded in the graph below, by the integration: $R_1=\int_0^{f(b)} \sqrt{y} \, dy = $\dfrac{2}{3}y^{\dfrac{3}{2}}$_0^{f(b)} = \, \dfrac{2}{3}f(b)^{\dfrac{3}{2}}\$ $R_2$ is not as straight forward because the two function do not intersect, so I'm not sure how to get the upper bound. I originally thought that if for some value 'b' they had the same $f(b)$ I could use that value as I drew in the graph below, but now I see that that's not possible because that would mean they are both the same function. I should be able to set up the integral with what I know from the question from the question even if I don't have the bounds. I know $y=x^2$ $x=0$ and $y=b^2$ If I solve $y=x^2$ and $y=b^2$ for $x$ $y=x^2$ is $x=\sqrt{y}$ but I dont think I can say $y=b^2$ -----> $b=\sqrt{y}$: At any rate if I put that together I would have $R_2=\int\sqrt{y}-\sqrt{y}\,dy=\int 0 \,dy$ which will obviously not work. Im a mess on this one, Im going to have to think about it.
 October 15th, 2011, 06:48 AM #5 Member   Joined: Oct 2011 Posts: 45 Thanks: 0 Re: Volumes By Cylindrical Shells the changing of the number sequence may be the key
 October 15th, 2011, 09:17 AM #6 Senior Member   Joined: Sep 2011 From: New York, NY Posts: 333 Thanks: 0 Re: Volumes By Cylindrical Shells I don't know what you mean.
 October 16th, 2011, 12:49 PM #7 Member   Joined: Oct 2011 Posts: 45 Thanks: 0 Re: Volumes By Cylindrical Shells number sequences get it the positioning of the numerals got it
October 16th, 2011, 12:53 PM   #8
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Re: Volumes By Cylindrical Shells

Quote:
 Originally Posted by mr.math number sequences get it the positioning of the numerals got it
Rather than respond with at the very least an impatient tone, why not try to explain what you mean not only for the benefit of the OP, but for all that follow who read this thread and benefit too? That's a forum at its best, clearly explained ideas on various topics for all to see.

October 20th, 2011, 05:09 AM   #9
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Re: Volumes By Cylindrical Shells

Quote:
Originally Posted by MarkFL
Quote:
 Originally Posted by mr.math number sequences get it the positioning of the numerals got it
Rather than respond with at the very least an impatient tone, why not try to explain what you mean not only for the benefit of the OP, but for all that follow who read this thread and benefit too? That's a forum at its best, clearly explained ideas on various topics for all to see.
i agree that is what i am trying to do i am trying to get my numerical points across by using my formula

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