Volume formula for cap of sphere

• Sep 11th 2006, 11:52 AM
Volume formula for cap of sphere
A hemispherical bowl of radius r contains water to a depth height h. Give a formula that you can use to measure the volume of the water in the bowl.
• Sep 11th 2006, 12:27 PM
ThePerfectHacker
I would use integration consider a semi-circle.
$\displaystyle y=\sqrt{r^2-x^2}$.
Let $\displaystyle h$ be the height i.e. distance from the endpoint on semicircle. Then what distance from the origon is $\displaystyle r-h$.
Thus, the volume by revolving about the x-axis is given by,
$\displaystyle \pi \int_{r-h}^r r^2-x^2dx$

Note: LaTeX (equation editor) is currectly disabled. In a few days these strange symbols I wrote down should turn into equations. If you know LaTeX you can follow this still.
• Sep 11th 2006, 12:29 PM
galactus
There are various ways, but you could come up with a formula by rotating about the y-axis.

$\displaystyle {\pi}\int_{-r}^{h-r}(r^{2}-y^{2})dy=\frac{h^{2}{\pi}(3r-h)}{3}$

Try it with a sphere of radius 4, completely filled:

$\displaystyle {\pi}\int_{-4}^{4}(16-y^{2})dy=\frac{256{\pi}}{3}$

Check this against the formula for sphere volume.
• Sep 11th 2006, 05:39 PM
Quick