Let $f(x) = x^2$, and let P denote a partition of [a, b].
(b) Use the fact that $f(x)$ is continuous to show that for any $\epsilon > 0$ there exists $\delta > 0$ such that for all partitions P with $\Vert P \Vert < \delta$ and any choice of $x^*_i \in [x_{i-1}, x_i]$ $\vert R(f, P, x^*_i) - \frac{b^3-a^3}3 \vert < \epsilon$. Conclude that $\int_a^b x^2 dx = \frac{b^3-a^3}3$.
Upon my first attempt to answer the problem, I was marked wrong and sent for corrections. The advice I was given was to use continuity along with part (a) to answer part (b).
My initial thoughts were to state that $f(x)$ is in fact uniformly continuous over $[a,b]$ and hence is Riemann integrable. Then, using the fact that for any choice of test points of the partition, the Riemann integral is the same, I concluded that $R(f,P) = M(f,P) = L(f,P)$. However, this does not seem to answer the part of the question regarding the epsilon-delta proof. I would like to know if my answer suffices, or how I could relate what I have already shown to the part in question. Thank you.
NOTE: This question is related to, but not found in, the book Elementary Classical Analysis, 2nd ed. by Marsden and Hoffman.