There is a formula for this. What we have is two spherical caps.
The formula for the volume of a spherical cap is
$\displaystyle V=\frac{\pi}{3}h^{2}(3R-h)$
h=the height of the cap. In this case it is 3/2.
$\displaystyle V=\frac{\pi}{3}(\frac{3}{2})^{2}(3(3)-\frac{3}{2})=\frac{45{\pi}}{8}$
Since we have two of them, we multiply by 2 and get $\displaystyle \frac{45\pi}{4}$
Also, there is a general formula for the volume of two intersecting spheres.
$\displaystyle V=\frac{1}{12}{\pi}(4R+d)(2R-d)^{2}$
Where d is the distance between the radii, in this case 3.
$\displaystyle \frac{\pi}{12}(4(3)+3)(2(3)-3)^{2}=\frac{45\pi}{4}$
You could derive this using calcarooney if you wish, but when we have a formula, why bother?.
Way too complicated?. All you have to do is plug your values into the formula. It don't get easier than that.
We could derive the spherical cap volume formula using calc.
$\displaystyle {\pi}\int_{r-h}^{r}(r^{2}-y^{2})dy$
$\displaystyle =\left[r^{2}y-\frac{y^{3}}{3}\right]_{r-h}^{r}=\frac{\pi}{3}h^{2}(3r-h)$
There it is. When two spheres intersect, we have two caps so multiply by 2.
The fact that the spheres have the same radius makes it easier.