# Volume of two spheres using triple integrals

The Volume of a Sphere – Numberphile
The Volume of a Sphere – Numberphile

Lets have two spheres, the middle point of sphere 1 is on the edge of sphere2(see picture). If I want to calculate the volume that is inside this region of the two spheres, do I need to use cylindrical coordinates or spherical? And can someone show me how to find the boundaries

• 1$\begingroup$ Welcome to MSE. With spheres, it’s usually more convenient to use spherical coordinates than cylindrical (as the latter name implies, it’s more often used with cylinder type objects). As for finding the boundaries, you equate the equations for the $2$ spheres to see where the co-ordinates are equal. $\endgroup$ Aug 11, 2019 at 9:36

The volume that is shared by the two spheres is a volume of revolution which could be found by a single integral.

Note that the equation of the right hand side sphere is $$(x-1)^2+y^2=4$$

The section of that sphere which is in the second and the third quadrant is $$\int _{-1}^0 \pi y^2=\int _{-1}^0 \pi [4-(x-1)^2]dx =5\pi /3$$

Thus the total volume is twice that which is $$V=\frac {10\pi}{3}$$

• $\begingroup$ Thanks! this is correct but if I need to solve it with triple integrals I don’t see how we do the boundaries $\endgroup$– Peter17Aug 11, 2019 at 10:18
• $\begingroup$ Then you need to project on $yz$ plane and find the volume between the right and left surfaces on the circle $$y^2+z^2=3$$ using cylindrical coordinates. $\endgroup$ Aug 11, 2019 at 10:23
• $\begingroup$ I think I get it thanks! $\endgroup$– Peter17Aug 11, 2019 at 10:30

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