How Is the Volume of a Spherical Segment Calculated Using Cavalieri's Principle?

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Homework Statement

Use Cavalieri's Principle to find the volume of a spherical segment of one base and thinckness h if the radius of the sphere is r.

Homework Equations

http://img395.imageshack.us/img395/2826/sphere1.jpg

Volume of half-sphere: 2/3$$\pi$$r²
Volume of cone inverse to half-sphere: 1/3 $$\pi$$r²

The Attempt at a Solution

I've been working this for the last three days and can't see how the answer is derived. The best I've been able to do is work out a cone with height and radius $$\alpha$$ where $$\alpha$$= r-h

But I haven't had any success this way, and question its usefulness.

If need be, I can post what the answer is supposed to be, I'm just interested in how its derived.

 

[larstuff]

Homework Statement

Use Cavalieri's Principle to find the volume of a spherical segment of one base and thinckness h if the radius of the sphere is r.

Homework Equations

http://img395.imageshack.us/img395/2826/sphere1.jpg

Volume of half-sphere: 2/3$$\pi$$r²
Volume of cone inverse to half-sphere: 1/3 $$\pi$$r²

The Attempt at a Solution

I've been working this for the last three days and can't see how the answer is derived. The best I've been able to do is work out a cone with height and radius $$\alpha$$ where $$\alpha$$= r-h

But I haven't had any success this way, and question its usefulness.

If need be, I can post what the answer is supposed to be, I'm just interested in how its derived.

Le Cavalieris principle is about functions in the plane (2d) that are "hightened" into space (3d) by revolving them about the x-axis. A cut half circle would do here.

 

[Architeuthis Dux]

Hello,

Thank you for sharing your attempt at solving this problem. It seems like you have made a good start by using Cavalieri's Principle, which states that if two solids have the same height and their cross-sectional areas are equal at every height, then their volumes are equal.

To find the volume of a spherical segment, we can use a similar approach as finding the volume of a cone. The key is to find the cross-sectional area of the spherical segment at every height. Let's start with the cross-sectional area at the top of the segment, which is a circle with radius r. This is the same as the cross-sectional area of the half-sphere, which is 1/2 of the volume of the sphere. So, the volume of the top part of the spherical segment is 1/2 of the volume of the sphere, which is 2/3 πr^3.

Now, let's look at the cross-sectional area at the bottom of the segment, which is a circle with radius r-h. This is the same as the cross-sectional area of the cone inverse to the half-sphere, which is 1/3 of the volume of the half-sphere. So, the volume of the bottom part of the spherical segment is 1/3 of 2/3 πr^3, which is 2/9 πr^3.

To find the volume of the middle part of the segment, we can use Cavalieri's Principle. Imagine slicing the spherical segment into thin horizontal slices. At each height, the cross-sectional area is a circle with radius r - h. So, the volume of each slice is 1/3 of the volume of the cone inverse to the half-sphere, which is 1/9 πr^3. The height of each slice is dh, so the volume of the middle part of the segment is the integral of 1/9 πr^3 dh from h to r, which is 1/9 πr^3 (r-h).

Therefore, the total volume of the spherical segment is 2/3 πr^3 + 2/9 πr^3 + 1/9 πr^3 (r-h) = 1/3 πr^2h.

I hope this helps. Good luck with your studies!