Answer
\begin{align}
&(a) \ See \ solution. \\
& (b) \ Area = \frac{b^{4}}{4} - \frac{a^{4}}{4}
\end{align}
Work Step by Step
(a) We have to prove that the area under the graph y = $x^{3}$ over the interval [0, b] is $\frac{b^{4}}{4}$.
\begin{align}
Area = \int_{0}^{b}x^{3}dx = \big[\frac{x^{4}}{4}\big]_{0}^{b} = \frac{b^{4}}{4}
\end{align}
(b) Now, the given interval is [a, b] where a $\geq$ 0.
\begin{align}
Area = \int_{a}^{b}x^{3}dx = \big[\frac{x^{4}}{4}\big]_{a}^{b} = \frac{b^{4}}{4} - \frac{a^{4}}{4}
\end{align}
