Chapter 4.1, Problem 26E

 (a) Use Definition 2 to find an expression for the area under the curve $y=x^3$ from 0 to 1 as a limit.

(b) The following formula for the sum of the cubes of the first n integers is proved in Appendix E. Use it to evaluate the limit in part (a).

$1^3+2^3+3^3+\mathrm{...}+n^3={\left[\frac{n\left(n-1\right)}{2}\right]}^2$