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(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).

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

(A). $\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{i^{3}}{n^{4}}$

(B). $\frac{1}{4}$

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Campbell University

University of Nottingham

Idaho State University

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