I want a volume, i can't get it...

Well, i have a problem with a little integral. The fact is i have two similar circle crossing each other. I want to know the area of the crossing surface. it's easy it's πr²(Th - cos(Th))/2. (i think i do not have it in mind, it may seems like that however) Where Th is the half angle bewteen the two crossing point. And r the radius of course. So I have y surface, but i want a volume, so i tried to integrate it like a rectangle to have a cylinder but here i'll have something like a rugby ball (or football for US bois :) ) But i can't make, will you guys have an idea on this ? How do i integrate my aera to have a volume ?