My Math Forum Volume of intersection of cylinders

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 November 30th, 2012, 12:56 PM #1 Member   Joined: Nov 2012 Posts: 50 Thanks: 0 Volume of intersection of cylinders The axes of two right circular cylinders of radius æ interstect at a right angle. Find the volume of the solid of intersection of the cylinders. Why radius is æ, well, I don't know.
 November 30th, 2012, 01:01 PM #2 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,155 Thanks: 463 Math Focus: Calculus/ODEs Re: Volume of intersection of cylinders We can use slicing to find the volume of the resulting solid. If we choose an axis z perpendicular to the two axes of symmetry of the intersecting cylinders of radii r, we find square cross sections, whose side lengths s are $\sqrt{r^2-z^2}$, thus the volume V is: $V=4\int_{-r}\,^r r^2-z^2\,dz=8\int_{0}\,^r r^2-z^2\,dz=8$r^2z-\frac{z^3}{3}$_0^r=8$$r^3-\frac{r^3}{3}$$=8r^3$$1-\frac{1}{3}$$=\frac{16}{3}r^3$
 November 30th, 2012, 02:27 PM #3 Member   Joined: Nov 2012 Posts: 50 Thanks: 0 Re: Volume of intersection of cylinders Answer is correct. I understood that you wrote all stuff to get correct answer. But thanks for explanation. And could you solve this problem? http://mymathforum.com/viewtopic.php?f=13&t=36334
November 30th, 2012, 04:02 PM   #4
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Re: Volume of intersection of cylinders

Quote:
 Originally Posted by etidhor And could you solve this problem? http://mymathforum.com/viewtopic.php?f=13&t=36334
Sorry, homework not done here.
Please show your work: we can then tell you where you're wrong. Thank you.

December 1st, 2012, 01:24 AM   #5
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Re: Volume of intersection of cylinders

Quote:
 Originally Posted by MarkFL We can use slicing to find the volume of the resulting solid. If we choose an axis z perpendicular to the two axes of symmetry of the intersecting cylinders of radii r, we find square cross sections, whose side lengths s are $\sqrt{r^2-z^2}$, thus the volume V is: $V=4\int_{-r}\,^r r^2-z^2\,dz=8\int_{0}\,^r r^2-z^2\,dz=8$r^2z-\frac{z^3}{3}$_0^r=8$$r^3-\frac{r^3}{3}$$=8r^3$$1-\frac{1}{3}$$=\frac{16}{3}r^3$
Where is pi??? I think u solution is incorrect, yesterday while i was in bed i solved this problem in my brain and answer had pi in it

 December 1st, 2012, 01:39 AM #6 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,155 Thanks: 463 Math Focus: Calculus/ODEs Re: Volume of intersection of cylinders Since the cross-sections are square, there is no pi. The solution I gave is correct.
 December 1st, 2012, 02:17 AM #7 Member   Joined: Nov 2012 Posts: 50 Thanks: 0 Re: Volume of intersection of cylinders Dude it can't be square, the intersection will be spherical cube, 'I couldn't remember exact word for it' just use math software to draw it, and you will see.
 December 1st, 2012, 02:20 AM #8 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,155 Thanks: 463 Math Focus: Calculus/ODEs Re: Volume of intersection of cylinders I googled it, and found these: http://www.math.tamu.edu/~tkiffe/cal...2cylinder.html http://mathworld.wolfram.com/SteinmetzSolid.html
December 1st, 2012, 05:46 AM   #9
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Re: Volume of intersection of cylinders

Quote:
 Originally Posted by MarkFL I googled it, and found these: http://www.math.tamu.edu/~tkiffe/cal...2cylinder.html http://mathworld.wolfram.com/SteinmetzSolid.html
Sorry, you are right, I got it, I just don't believe anything unless solid proof is brought.

 December 1st, 2012, 08:44 AM #10 Member   Joined: Nov 2012 Posts: 50 Thanks: 0 Re: Volume of intersection of cylinders Now can anyone explain how did Archimedes calculated it without calculus? ...

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