Chapter 5.1, Problem 24E

(a) Use Definition 2 to express 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).

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