A volume?

**arbolis** · July 12th 2009, 09:25 PM

Here's the problem :
Let $B_4=\{ (x,y,z,w) \in \mathbb{R}^4 : x^2+y^2+z^2+w^2 \leq 1 \}$ and $B_3=\{ (x,y,z) \in \mathbb{R}^3 : x^2+y^2+z^2 \leq1 \}$ .
1)Show by a justification that $V(B_4)=2 \iiint _{B_3} \sqrt{1-(x^2+y^2+z^2)}dxdydz$ .
2)Deduce that $V(B_4)=\frac{\pi ^2}{2}$ .
My attempt : almost none. My first problem is : they never defined the function $V$ . 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 $B_3$ is the projection of $B_4$ in the 3 dimensional space.
Also, I recognize the integrand to be the positive $w$ that satisfy the first inequation. So $V(B_4)$ 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 $-w$ . 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.

**Chris L T521** · July 12th 2009, 09:47 PM

Originally Posted by arbolis

Here's the problem :
Let $B_4=\{ (x,y,z,w) \in \mathbb{R}^4 : x^2+y^2+z^2+w^2 \leq 1 \}$ and $B_3=\{ (x,y,z) \in \mathbb{R}^3 : x^2+y^2+z^2 \leq1 \}$ .
1)Show by a justification that $V(B_4)=2 \iiint _{B_3} \sqrt{1-(x^2+y^2+z^2)}dxdydz$ .
2)Deduce that $V(B_4)=\frac{\pi ^2}{2}$ .
My attempt : almost none. My first problem is : they never defined the function $V$ . 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 $B_3$ is the projection of $B_4$ in the 3 dimensional space.
Also, I recognize the integrand to be the positive $w$ that satisfy the first inequation. So $V(B_4)$ 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 $-w$ . 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.

$x^2+y^2+z^2+w^2=1$ is a ball in 4-space (or a 4-D sphere).

If we are to find the volume, we solve for the function w(x,y,z). In this case, $w=\sqrt{1-x^2-y^2-z^2}$ . Recall that when we find volumes in 3-space, we resort to a region in 2-space to determine the limits. In our case, we resort to a region in 3-space in order to find our limits. Then we can find the volume of the ball in 4-space.

Thus, it would make sense that $V\left(B_4\right)=2\iiint\limits_{B_3}\sqrt{1-x^2-y^2-z^2}\,dz\,dy\,dx$ (we need two triple integrals to take into consideration the two hemispheres of the ball).

From here, evaluate the integral. A simple change to spherical coordinates will do the job.

Can you continue? I hope my explanation makes sense...

**arbolis** · July 13th 2009, 07:31 AM

Ok thanks Chris, I'll try to do it. I was accustomed to speak about volumes only for solids in 3 dimensions. Now I realize that a 3 dimensional sphere has an area ( $4\pi r^2$ ) and with the same idea, a 4 dimensional sphere has a volume!
I'm curious, does a 4 dimensional sphere also has an area?

**Danny** · July 13th 2009, 03:02 PM

Originally Posted by arbolis

Ok thanks Chris, I'll try to do it. I was accustomed to speak about volumes only for solids in 3 dimensions. Now I realize that a 3 dimensional sphere has an area ( $4\pi r^2$ ) and with the same idea, a 4 dimensional sphere has a volume!
I'm curious, does a 4 dimensional sphere also has an area?

2D surface area (i.e. arc length) $\int_R \sqrt{1+y'^2} dx$ in 3D $\iint_R \sqrt{1+z_x^2+z_y^2} \,dA$

What do you think?

**arbolis** · July 13th 2009, 04:10 PM

Originally Posted by Danny

2D surface area (i.e. arc length) $\int_R \sqrt{1+y'^2} dx$ in 3D $\iint_R \sqrt{1+z_x^2+z_y^2} \,dA$

What do you think?

Seems to make sense. Did you deduce it, or saw it in a book/internet?

**Danny** · July 14th 2009, 04:54 AM

Originally Posted by arbolis

Seems to make sense. Did you deduce it, or saw it in a book/internet?

Here's an interesting link

Hypersphere -- from Wolfram MathWorld

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