volume of the cap of a sphere

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mares09

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Jul 11, 2007
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I have been trying to work on this problem but I cant figure it out.

Using integration techniques find the formula for the volume of the cap of a sphere with radius R. The cap has a height, H.

So far sliced the cap in two parts and I know that i have to rotate it around the x axis. I know if i let the bottom of the cap be x, then in x = square root of (R^2 - (R-H)^2). Thats all I have been able to figure out.

Thank you for your help!!
 
mares09 said:
I have been trying to work on this problem but I cant figure it out.

Using integration techniques find the formula for the volume of the cap of a sphere with radius R. The cap has a height, H.

So far sliced the cap in two parts and I know that i have to rotate it around the x axis. I know if i let the bottom of the cap be x, then in x = square root of (R^2 - (R-H)^2). Thats all I have been able to figure out.

Thank you for your help!!
Click to expand...

This is a typical application of the disc method.

Draw a sketch first.

Now a disk at the distance 'x' would have radius of 'y' (= \(\displaystyle sqrt{R^2 - x^2}\))

The volume of the disk is then dv = pi * y^2 dx

Now integrate over the limit you have already found.
 
I was getting v= (antiderivative) pi (R^2 - X^2).

the limit is from (r-h) to h.

Im confused is there any way that you can post a graph with the equations? Thank you.
 
sphericalcap1001kt8.png


\(\displaystyle \L\\{\pi}\int_{r-h}^{r}(r^{2}-x^{2})dx\)
 
Thank you galactus. I tried to integrate the equation and I get:

((R^3) / 3) - ((X^3) / 3) then I evaluate for the limits and I get

-( (R^3) - ((R^3) - (-3(R^2)h) + (3R(h^2) - h^3) / 3

This would be (3R(h^2) - 3 (R^2)h - (h^3)) / 3

and then the answer should be times pi. The problem is that I know the answer should be pi ((h^2) (3R-h)) /3.

what am I doing wrong?[/img]
 
Let's integrate it then:

r is a constant. You're integrating it like x. The integral of r^2 in this case is x*r^2

\(\displaystyle \L\\\int_{r-h}^{r}(r^{2}-x^{2})dx=r^{2}x-\frac{x^{3}}{3}|_{r-h}^{r}\)

\(\displaystyle \L\\\left[r^{2}(r)-\frac{(r)^{3}}{3}\right]-\left[r^{2}(r-h)-\frac{(r-h)^{3}}{3}\right]\)

\(\displaystyle \L\\\left(\frac{2r^{3}}{3}\right)-\left(\frac{2r^{3}}{3}-h^{2}r+\frac{h^{3}}{3}\right)=h^{2}r-\frac{h^{3}}{3}\)

Don't forget PI. Factor:

\(\displaystyle \L\\\fbox{\frac{h^{2}{\pi}(3r-h)}{3}}\)
 
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