On my Calculus homework, there is a question that I am having trouble with.
The question is:
Rewrite
$\lim_{n \to \infty}\sum_{i=1}^{n}\frac{1}{n}\left[\left(\frac{i}{n}\right)^{3}+\left(\frac{i}{n}\right)^{3}+\ldots+\left(\frac{n-1}{n}\right)^3\right]$ as an integral $\int_{a}^{b}f(x)dx$
I know how to do this kind of problems, but this one is a little confusing.
I know that:
$$\Delta x = \frac{b-a}{n}\\x_i = \Delta x i + a\\\lim_{n \to \infty}\sum_{i=1}^{n}f(x_i)\Delta x=\int_{a}^{b}f(x)dx$$
and in the question
$$\begin{eqnarray*}\frac{1}{n} &=& \frac{b-a}{n} = \Delta x\\1&=&b-a\\a&=&0\\b&=&1\end{eqnarray*}$$
So the answer must be something like:
$$\int_{0}^{1}f(x)dx$$
What I don't understand is the $f(x_i)$ part.
I don't know how to simplify the expression inside the bracket.
It becomes something like:
$$\left[x^{3}+x^{3}+\ldots+\left(\frac{n-1}{n}\right)^{3}\right]$$
I can't figure out what $f(x)$ is.
Any help or explanations are appreciated, thanks.
Edit:
The question in my homework might have a typo. Thank you for your help.
Best Answer
Recall that we are evaluate rectangular area $f(x_i)\cdot \Delta x$, here $x_i = \frac{i}{n}$ and then we sum up the rectangular areas.
$$f(x)=x^3$$