Problem 1
Consider the function defined by $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are real constants.
(a) For the specific case $a = \frac{1}{2}$, $b = -3$, $c = 4$, evaluate the right-endpoint Riemann sum for
the function $f$ over the interval $[1, 3]$ using a regular partition with 4 subintervals.
The remaining parts of this problem refer to the general case with unspecified values of
$a$, $b$, and $c$.
(b) Explain briefly how you know that $f$ is integrable on the interval $[1, 3]$.
(c) Find an expression for the left-endpoint Riemann sum of $f$ over the interval $[1, 3]$ using
regular partitions with an arbitrary natural number $n$ of subintervals.
(d) Express $I = \int_1^3 f(x)dx$ as a limit of the Riemann sums in (c) and evaluate this limit to
find a formula for $I$ in terms of $a$, $b$, and $c$.
