Volume of a spere with 2 sections taken out of it

[Biftheunderstudy]

Hey guys I'm working on a project and I've run into a snag and I haven't done vector calculus in years. Here's the problem:

Picture a sphere partially filled with liquid up to some height, now slice the sphere in the vertical plane at some depth. Effectively a spherical cap oriented vertically filled partially. I need to know the volume of that partially filled cap.

Thanks

 

[slugg]

Can you draw a picture or something? I'm not understanding the shape...

 

[masteryoda34]

Originally posted by: slugg
Can you draw a picture or something? I'm not understanding the shape...

Me neither.

 

[silverpig]

I stole this from another forum:

For a sphere with radius = r

Let h be the height of liquid in the sphere.

If we subdivide the height into thin segments by dh, let x be the radius at height h.

So when h is 0, then x is 0 (empty sphere)
and when h is r, then x is r (1/2 full sphere {known to be 2/3 PH r^3}

We know that the circumfrential area for any little dh, is CA = PI * x^2

We need the relationship between x and h.

As h increases, the remaining distance from the center of the sphere to the top of the liquid is r-h.

The length of the line from the center of the sphere to the edge of the liquid is r

So we can find x using x^2 + (r-h)^2 = r^2
x^2 = r^2 - {r^2 -hr -hr + h^2}
x^2 = r^2 - {r^2 -2hr + h^2}
x^2 = r^2 - r^2 + 2hr - h^2
x^2 = + 2hr - h^2
x = {2hr -h^2}^(1/2)


To find the volume of liquid in the sphere

Volume = int (PI x^2) dh from 0 to h
V = PI int (2hr - h^2) dh from 0 to h
V = PI { int (2rh) dh - int (h^2) dh from 0 to h
V = PI { (rh^2) - ((1/3) h^3) } from 0 to h


so when h is 0, V is 0 as both terms collapse

when h is r, V is
V = PI { (r^3) - ((1/3) r^3) } from 0 to h
V = (2/3) PI { (r^3)

and when h is 2r, V is
V = PI { (rh^2) - ((1/3) h^3) } from 0 to h
V = PI { (4r^3) - ((8/3) r^3) }
V = (1/3) PI { (12-8) r^3}
V = (4/3) PI r^3

What you want is the equation:

V = pi { (rh^2) - ((1/3) h^3) }

h is in terms of r. So for a half-filled sphere you set h = r.

So that's the first part of the question. To make it easy divide your answer by 4/3 pi r^3 to get it in terms of a fraction of the total volume. You should now have some fraction A.

To do the second part, just do the same thing to get some fraction B. Your total volume is then A * B * V.

 

[Biftheunderstudy]

Awesome, I had it right up to calculating the 2 volumes. Couldn't remember what to do with them though.
Thanks!