Finding the area under a graph using Riemann Sum question

I would appreciate any help I can get with this problem! My prof usually gives us a few practice problems we should know. One of them this week was finding the area under the graph y=x^3 using Riemann Sum. Even in high school, I have had a hard time grasping how to find the area using Riemann Sum. I do not think it was explained to me right the first time.

**Problem: find the area underneath the graph y=x^3 using eight subdivisions and rectangles underneath y.**

The range is from 0 to 1

in my prof's example (of y=x^2) he used the following information:

the area of each rectangle is i^2/n^3 where i=(n-1)

so: [0^2]/n^3 + [1^2]/n^3 + [2^2]/n^3.... [(n-1)^2]/n^3

and because: [m(m+1)]/2=1+2+3...m

and 1^2+2^2...m^2=m(m+1)(2m+1)/6

then: [0^2]/n^3 + [1^2]/n^3 + [2^2]/n^3.... [(n-1)^2]/n^3 = (n-1)(n)2n-1)/

I know that the above is confusing (at least in IMO) but I really would appreciate any guidance at all. Let me know if left any information out or was not clear enough