Question

A question on n-dimensional volume integrals . See picture below.

Answer 1

\[2 \times \int\limits_{P} d(x3,x4,...,xn) where P=[a3,b3]\times...\times[an,bn] . \] I would like to know if the notation I chose is correct? Thanks king !

Answer 2

Well since you're integrating over a "rectangle" in \(n\)-dimensional space you can integrate in whatever order you want. So you can rewrite your integral as\[\int_0^nx_n\int_0^{n-1}x_{n-1}\dots\int_0^2 x_2\int_0^1 x_1\,dx_1dx_2...dx_{n-1}dx_n\]

Answer 3

Now we can just individually evaluate each integral. The innermost one is \(\int_0^1x\,dx=\frac{1}{2}\). This is a constant so you can pull it out of the whole integral. The next integral you evaluate is \(\int_0^2 x\,dx=2\). Again this is a constant we can pull out of the integral. So now we have\[\frac12\cdot2\int_0^nx_n\dots\int_0^3x_3\,dx_3\dots dx_n.\]

Answer 4

Long story short, the end result will be\[\left[\int_0^1x_1\,dx_1\right]\dots\left[\int_0^nx_n\,dx_n\right].\]So just evaluate each integral and multiply them all together, and you're done.

Answer 5

I actually messed up the simple integration of x_1 and x_2 , which I should be ashamed of ahaha, thanks KING !

Answer 6

We all make silly mistakes sometimes :P

And I will say that the notation you used is perfectly fine in this case. Assuming you integrated correctly of course :P