Question

Can anyone help with using limit of Reimann sum?

Answer 1

\[\int\limits_{0}^{1}(x^2 + x + 1)dx\]

Answer 2

I know there is a much easier way to find area under a curve, but I have to be able to do it this way also for test

Answer 3

I know \(\Delta x\) is 1/n

Answer 4

And \(x_k\) is k/n

Answer 5

Not sure what to do from there

Answer 6

Let's make the problem a bit more general for a moment.

You're trying to approximate the integral of some function \(y=f(x)\) over an interval \([a,b]\). You do so by partitioning the interval into \(n\) subintervals, so that the integral/area is approximated by \(n\) rectangles with width \(\Delta x=\dfrac{b-a}{n}\), and according to your comments above, height \(f(x_k)=f\left(\dfrac{k}{n}\right)\). This corresponds to a left-endpoint approximation.

The integral is then approximated by
\[\int_a^bf(x)\,\mathrm dx\approx\sum_{k=0}^{n+1}f\left(\frac{k}{n}\right)\Delta x\]Now back to your problem. It's clear enough that \(\Delta x=\dfrac{1-0}{n}=\dfrac{1}{n}\), and that
\[\begin{align*}
f(x)=x^2+x+1\implies f\left(\dfrac{k}{n}\right)&=\left(\dfrac{k}{n}\right)^2+\dfrac{k}{n}+1\\[1ex]
&=\frac{k^2}{n^2}+\frac{k}{n}+1\\[1ex]
&=\frac{k^2+kn+n^2}{n^2}
\end{align*}\]So, the integral is approximately given by
\[\int_0^1(x^2+x+1)\,\mathrm dx\approx\sum_{k=0}^{n+1}\frac{k^2+kn+n^2}{n^2}\times\frac{1}{n}=\frac{1}{n^3}\sum_{k=0}^{n+1}(k^2+kn+n^2)\]Do you know how to proceed from here?