# Thread: Cyndrical shells for volume of a sphere

1. ## Cyndrical shells for volume of a sphere

I have to use the cylindrical shells formula to find the volume of a sphere of radius r. I wasn't sure on how to solve it, so I checked my answer guide for help. I'm confused on: how did the positive 2s from the first step turn into negative 2s in the second step? And on the third step, the (-2x) disappeared? What was solved to make these steps possible?

2. Originally Posted by !!!
I have to use the cylindrical shells formula to find the volume of a sphere of radius r. I wasn't sure on how to solve it, so I checked my answer guide for help. I'm confused on: how did the positive 2s from the first step turn into negative 2s in the second step? And on the third step, the (-2x) disappeared? What was solved to make these steps possible?

That is because the derivative of the (r^2 -x^2) is -2x dx
You have only x dx in the original integral, so, in order for you to differentiate the integrand, the x dx must become -2x dx.
So you multiply and divide at the same time the x dx by -2.

In the integration,
V = (-4pi/3)[(r^2 -x^2)^(3/2)]|(0 to r),
V = (-4pi/3)[(r^2 -r^2)^(3/2) -(r^2 -0)^(3/2)]
V = (-4pi/3)[(0) -(r^2)^(3/2)]
V = (-4pi/3)[-r^3]
And so the negatives disappeared again.

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### derive the volume of a sphere using cylindrical shells

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