Originally Posted by

**arbolis** Here's the problem :

Let $\displaystyle B_4=\{ (x,y,z,w) \in \mathbb{R}^4 : x^2+y^2+z^2+w^2 \leq 1 \}$ and $\displaystyle B_3=\{ (x,y,z) \in \mathbb{R}^3 : x^2+y^2+z^2 \leq1 \}$.

1)Show by a justification that $\displaystyle V(B_4)=2 \iiint _{B_3} \sqrt{1-(x^2+y^2+z^2)}dxdydz$.

2)Deduce that $\displaystyle V(B_4)=\frac{\pi ^2}{2}$.

My attempt : almost none. My first problem is : they never defined the function $\displaystyle V$. If I assume it's a volume function then they seem to ask me to calculate the volume of a 4 dimensional solid (or whatever it is called), which seems senseless. I realize that $\displaystyle B_3$ is the projection of $\displaystyle B_4$ in the 3 dimensional space.

Also, I recognize the integrand to be the positive $\displaystyle w$ that satisfy the first inequation. So $\displaystyle V(B_4)$ really seems a volume since there's the multiplication by 2 in front of the triple integral so that my instinct tells me that this covers the $\displaystyle -w$. I may not be clear here, but that's how I understand the problem.

If you understand it better than I, feel free to reformulate it so that I can understand it.