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# Calculus/Calculating the volume of R^3 lay between the sphere and cilinder

Question
Dear Abe,

So you have a 3d space where a sphere is present with the equation

x²+y²+z²=4a²

z=sqrt(4a²-x²-z²)

A cilinder cuts a certain volume out of this sphere, it's up to me to calculate this volume. The cilinder has cartesian coordinates

x²+y²=2ay

I can recognize the equation of a circle (x-x0)²+(y-y0)²=r²
I rewrite it in this form to find the center of the cilinder

x²+y²-2ay+a²=a²

(x-x0)²+(y-a)²=a²

so center is (0,a) and radius is a

I go onto polar coordinate system:

p²=2apsin&

p(max)=2asin&
p(min)=0

&(max)=pi
&(min)=0

So I rewrite the integral

2*integral(0 to pi)integral(0 to 2asin&)sqrt(4a²-p²)pdpd&
-1*integral(0 to 2pi)integral(0 to 2asin&)sqrt(4a²-p²)d(4a²-p²)d&

This gives me

-1*integral(0 to 2pi)[(4a²-p²)^(3/2)/(3/2)]from p=0 to p=2asin&

Up to this I know the calculations are correct. However the following things aren't very clear to me:

Why is that z=sqrt(4a²-x²-y²) only half of the volume is?
Why are the borders of the integral to & readjusted to 2pi during integration?

So now I need to integrate further to &, but there is where I get lost

The result I need to get is (16a^3/6)*(pi-(4/3))

Thanks,
Bob

Hello Bob,

z=sqrt(4a²-x²-y²) is only half of the volume since the cylinder and the sphere
also intersect on the other side (z<0) of the xy-plane.

Your first integral is correct: 2*integral(0 to pi)integral(0 to 2asin&)sqrt(4a²-p²)pdpd&
which gives: (16/9)*a^3*(3*Pi-4)...I think the "6" in your answer should be a "3"...

Abe

Calculus

Volunteer

#### Abe Mantell

##### Expertise

Hello, I am a college professor of mathematics and regularly teach all levels from elementary mathematics through differential equations, and would be happy to assist anyone with such questions!

##### Experience

Over 15 years teaching at the college level.

Organizations
NCTM, NYSMATYC, AMATYC, MAA, NYSUT, AFT.

Education/Credentials
B.S. in Mathematics from Rensselaer Polytechnic Institute
M.S. (and A.B.D.) in Applied Mathematics from SUNY @ Stony Brook