Question

Show all work and reasoning

Answer 1

Split up the interval [2, 5] into $n$ equally spaced subintervals, then consider the value of $f(x)$ at the right endpoint of each subinterval.

The length of the interval is $5-2=3$, so the length of each subinterval would be $\dfrac3n$. This means the first rectangle's height would be taken to be $x^2$ when $x=2+\dfrac3n$, so that the height is $\left(2+\dfrac3n\right)^2$, and its base would have length $\dfrac{3k}n$. So the area under $x^2$ over the first subinterval is $\left(2+\dfrac3n\right)^2\dfrac3n$.

Continuing in this fashion, the area under $x^2$ over the $k$th subinterval is approximated by $\left(2+\dfrac{3k}n\right)^2\dfrac{3k}n$, and so the Riemann approximation to the definite integral is

$\displaystyle\sum_{k=1}^n\left(2+\frac{3k}n\right)^2\frac{3k}n$

and its value is given exactly by taking $n\to\infty$. So the answer is D (and the value of the integral is exactly 39).