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August 21st, 2019, 12:11 AM   #1
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Calculating the volume under the intersection of three orthogonal cylinders by slices

I have three cylinders :
$\displaystyle x^2 + y^2 = 1 $
$\displaystyle y^2 + z^2 = 1 $
$\displaystyle x^2 + z^2 = 1 $

Now all these intersect orthogonally at the origin and I need to find the volume of than enclosure.
My problem is I'm not able to do it by slicing method. See what I'm trying to do is finding the intersection points first :-

$\displaystyle x^2 + y^2 = y^2 + z^2 $ since both equal to 1
therefore, $\displaystyle x = z$
Now, $\displaystyle 2 z^2 = 1 $ implies $\displaystyle z = plus/minus \sqrt{1/2} $
So, endpoints of the figure are $\displaystyle x = [-\sqrt {1/2} , \sqrt{1/2}] , y= -\sqrt{1/2}, \sqrt{1/2}], z = [-\sqrt{1/2}, -\sqrt{1/2}] $

Now I know that to find volume we have to integrate twice, but I can't figure out what we have to integrate, I have attached the figure. I want to know how can we proceed with slicing.

Thank you, any help will be much appreciated.
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Adesh Mishra is offline  
 
August 21st, 2019, 12:40 AM   #2
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August 21st, 2019, 04:49 AM   #3
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Wikipedia explains it for two cylinders but when it comes to three cylinders it just says “we use the same argument as before”. Can you please give a hint about slicing ?
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