Volume of a Solid

[Snowdog2112]

I need help finding "The volume of the solid that is bounded by two right circular cylinders of radius r, if their axis meet at angle θ.". I'm not sure how to get this one started. Any help would be appreciated, thank you.

 

[galactus]

This is sort of like the cylinders that meet at right angles, except the intersections do not create a square, but instead a 'diamond'.

The lengths of the sides of the square in the former problem was

$\displaystyle \sqrt{r^{2}-y^{2}}$, therefore the area of the square was

$\displaystyle r^{2}-y^{2}$

Now, we have to use a little trig to find the length of the side.

I wish I had a picture. Hey, I do have a picyure, albeit, hackneyed and comical. It's the best I can do with that infernal 'paint'. I hope it's enough to give you the idea.

Try:

\(\displaystyle \L\\8\int_{0}^{r}csc({\theta})(r^{2}-y^{2})dy\)

Also,

\(\displaystyle \L\\4\int_{-r}^{r}csc({\theta})(r^{2}-y^{2})dy\)

 

[Snowdog2112]

I think that I've got it, thank you for the help.