Limits of Riemann sums

Dividing $[0,\ 1]$ into $n$ equal parts gives $n$ rectangles of width $\frac{1}{n}$ and height $f\!\left(\frac{k}{n}\right)$. If $f(x)$ is continuous, the sum of their areas tends to the definite integral as $n\to\infty$.

To rewrite a sum, read $\frac{k}{n}$ as $x$ and $\frac{1}{n}$ as $dx$. If the factor $\frac{1}{n}$ is not out in front, factor it out first so that the rest depends only on $\frac{k}{n}$.

Examples

1. 01Write it as a definite integral.$\displaystyle \lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^{n}\left(\frac{k}{n}\right)^{2}$AnswerHide

   Answer$\displaystyle \int_0^1 x^{2}\,dx$

   Steps

   1. The $\frac{1}{n}$ is the strip width, i.e. $dx$, and $\frac{k}{n}$ plays the role of $x$
   2. As $k$ runs from $1$ to $n$, $\frac{k}{n}$ runs from $0$ to $1$; so the interval is $0$ to $1$
   3. $\displaystyle \lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^{n}\left(\frac{k}{n}\right)^{2}$$\displaystyle = \int_0^1 x^{2}\,dx$
2. 02Write it as a definite integral.$\displaystyle \lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^{n}\left(\frac{k}{n}\right)^{3}$AnswerHide

   Answer$\displaystyle \int_0^1 x^{3}\,dx$

   Steps

   1. The $\frac{1}{n}$ is the strip width, i.e. $dx$, and $\frac{k}{n}$ plays the role of $x$
   2. As $k$ runs from $1$ to $n$, $\frac{k}{n}$ runs from $0$ to $1$; so the interval is $0$ to $1$
   3. $\displaystyle \lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^{n}\left(\frac{k}{n}\right)^{3}$$\displaystyle = \int_0^1 x^{3}\,dx$
3. 03Write it as a definite integral.$\displaystyle \lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^{n}\left(\frac{k}{n}\right)^{1}$AnswerHide

   Answer$\displaystyle \int_0^1 x\,dx$
4. 04Write it as a definite integral.$\displaystyle \lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^{n}\frac{1}{1+\frac{k}{n}}$AnswerHide

   Answer$\displaystyle \int_0^1 \frac{1}{1+x}\,dx$
5. 05Write it as a definite integral.$\displaystyle \lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^{n}\sqrt{\frac{k}{n}}$AnswerHide

   Answer$\displaystyle \int_0^1 \sqrt{x}\,dx$
6. 06Write it as a definite integral.$\displaystyle \lim_{n\to\infty}\sum_{k=1}^{n}\frac{n}{n^{2}+k^{2}}$AnswerHide

   Answer$\displaystyle \int_0^1 \frac{1}{1+x^{2}}\,dx$

FAQ

Q1What if there is no $\frac{1}{n}$ in the sum?AnswerHide

A

Factor it out. For example, $\sum_{k=1}^{n}\frac{n}{n^2+k^2}=\frac{1}{n}\sum_{k=1}^{n}\frac{1}{1+\left(\frac{k}{n}\right)^2}$, so the limit is $\int_0^1\frac{1}{1+x^2}\,dx$.

Q2Why are the limits of integration 0 and 1?AnswerHide

A

As $k$ runs from $1$ to $n$, $\frac{k}{n}$ runs from $\frac{1}{n}$ to $1$, and as $n\to\infty$ these points fill the interval from $0$ to $1$. If $k$ runs from $1$ to $2n$ instead, the interval becomes $0$ to $2$.