Calculate the volume of $D=\{(x,y,z):x^2+y^2+z^2<1,ax+by+cz<d\}$

Question

Calculate the volume of $D=\{(x,y,z):x^2+y^2+z^2<1,ax+by+cz<d\}$
I've thought the with wolg a=b=0 and then try to calculate it but i think the integral this way becomes more difficult

Answer 1

The plane points in the $(a,b,c)$ direction so rotate it to make it the new $z$ such that

$$z < \frac{d}{\sqrt{a^2+b^2+c^2}}$$

to normalize so it is a pure rotation (i.e. the Jacobian is $1$). We have two cases. If $\sqrt{a^2+b^2+c^2} > d>0$ we get

$$V = \frac{4\pi}{3} - \int_0^{2\pi} \int_{\frac{d}{\sqrt{a^2+b^2+c^2}}}^1 \int_0^{\cos^{-1}\left(\frac{d}{\rho\sqrt{a^2+b^2+c^2}}\right)} \rho^2\sin\phi \:d\phi \:d\rho\:d\theta$$

and just the volume of the cap for $d<0$ but if the plane does not intersect the sphere we get the whole volume. Can you take it from here?

Answer 2

I would not regard this as an algebraic question, but a geometrical one: $x^2+y^2+z^2<1$ is a simple square, $ax+by+cz<d$ is a simple plane. Either they intersect or they don't.
In case they don't intersect, then the result is either the volume of the unity sphere, or it is zero.
In case they intersect, you can use the basic geometry formula for the volume of a sphere segment or sector.

By the way: who or what is Fubini?