Finding the volume of this intersectionHow to find the volume of this:The region common to the interiors of the cylinders x 2 + y 2 = 1 and x 2 + z 2 = 1 and the first octant.I tried finding the volume via double integration and the integration gets too complicated. I want to know how to solve this via triple integration. Usually there are x, y, z′s in both equations and that I can solve with triple but I dont know what to do here. I started of doing x 2 + z 2 = x 2 + y 2 so z = y but dont know what to do with this it doesnt look like the line of intersection is z = y in the diagram given.

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Finding the volume of this intersectionHow to find the volume of this:The region common to the interiors of the cylinders       x    2    +      y    2    =  1 and       x    2    +      z    2    =  1 and the first octant.I tried finding the volume via double integration and the integration gets too complicated. I want to know how to solve this via triple integration. Usually there are x, y, z′s in both equations and that I can solve with triple but I dont know what to do here. I started of doing       x    2    +      z    2    =      x    2    +      y    2   so   z  =  y but dont know what to do with this it doesnt look like the line of intersection is   z  =  y in the diagram given.

nedervdq3 · Accepted Answer

Step 1Draw out the surfaces and see that x varies from 0 to 1, y varies from 0 to       1    −          x      2      , and z varies from 0 to       1    −          x      2       as well. There are 5 other ways to order the integration but aren&#039;t as useful as this ordering (check why) and this ordering makes use of squaring the square root   V  =      ∫    0    1        ∫    0          1      −              x        2                  ∫    0          1      −              x        2              d  z  d  y  d  x.Step 2This is the same as   V  =      ∫    0    1        ∫    0          1      −              x        2                  1    −          x      2        d  y  d  x.Which is the area integral of the function   f  (  x  ,  y  )  =      1    −          z      2      , the surface of the &quot;top&quot; cylinder. The other cylinder,       x    2    +      y    2    =  1 serves as the boundary of the integration region. The triple and double integrals are equivalent.When using the triple integral to find volume, the integrand is simply 1 and integrate wrt all variables whereas with using the double integral to find volume, the integrand is the function f(x,y) and then integrate wrt x and yAs for   z  =  y, it appears on the   y  −  z plane as the cylinders make contact at (0, 1, 1)

makeupwn · Accepted Answer

Explanation:Let A be the intersection of the two solid cylinders, considered only in the first octant   x  ,  y  ,  z  ≥  0.Use Fubini for the computation of the volume. It is clear that   x  ∈  [  0  ,  1  ]. Now for a fixed x we are searching for all (y,z) with the property   0  ≤  y  ,  z  ≤      1    −          x      2      . This is a square with side       1    −          x      2      , its area is   (  1  −      x    2    ) so the needed volume is   Volume  (  A  )  =      ∫    0    1    (  1  −      x    2    )    d  x  =  1  −      1    3    =      2    3       .

The region common to the interiors of the cylinders x^{2}+y^{2}=1 and x^{2}+z^{2}=1 and the first octant.

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