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\noindent Problems esergel52, esergel53, and esergel54 are intended to be used together.

\begin{questions}

\question  % workshop bank Ch. 5 -> esergel53
Recall that $\int_0^b x^2 \, dx = \frac{b^3}{3}$. In the bonus question after this, you will justify this formula for $b>0$ using Riemann sums. For now, assume that this is true when $b>0$.

\begin{parts}

\part How does $\int_0^5 x^2 \, dx$ relate to $\int_{-5}^{0} x^2 \, dx$? Draw a picture to illustrate your ideas.

\part What is $\int_{0}^{-5} x^2 \, dx$? What is the general formula for $\int_0^b x^2 \, dx$ when $b<0$?

\part Use properties of integrals to find a formula for $\int_a^b x^2 \, dx$ when $0<a<b$.

\part Justify that your formula from part (c) also works when $a<0<b$.

\end{parts}

\end{questions}

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