How do I find the volume cut off of the unit ball by the plane ax+by+cz = d? How do I find the volume cut off of the unit ball by the plane ax+by+cz = d?
I know that there's a double integral somewhere here, but I just don't understand how to attack this problem. Any guidance would be appreciated!

By symmetry of the unit ball (centered at $(0,0,0)$ I suppose), you may as well rotated the plane until it is parallel to the $xy$-plane. This can be done by first computing the distance $\lambda$ from $(0,0,0)$ to the plane and then we have the point set
$$
CAP=\{(x,y,z)\mid z\geq \lambda,\ y^2\leq 1-z^2,x^2\leq 1-y^2-z^2\}
$$
One fourth of this spherical cap has $x,y\geq 0$ and its volume can be calculated as
$$
\frac14 CAP=\int_\lambda^1 \int_0^{\sqrt{1-z^2}}\int_{0}^{\sqrt{1-y^2-z^2}}1\ dx\ dy\ dz
$$
