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\begin{document}

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

\begin{questions}

\question % workshop bank Ch. 5 -> esergel52

\begin{parts}

\part Use geometry to justify the formula $\int_0^b x \, dx = \frac{b^2}{2}$ when $b>0$.

\part Justify this formula when $b<0$. Remember that
$
\int_a^b f(x) \, dx = -\int_b^a f(x) \, dx.
$

\part What is the area under the curve $f(x)=x$ between $x=a$ and $x=b$ when $0<a<b$?

\part Check that this same formula also works when $a<0<b$. Draw a picture to illustrate your ideas.

\part Use geometry to evaluate the integrals $\int_{-1}^3 7x \, dx$ and $\int_{-2}^5 \left( 3x+6 \right) \, dx$. Draw pictures of the two areas being computed. Check your answers against the formula from part (d). Use the properties of integrals discussed in class, and remember that $\int_a^b K \, dx = K(b-a)$.

\end{parts}

\end{questions}

\end{document}
