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Prove that if $f(x)$ is integrable between $0$ and $1$ , then $\begin{equation} \lim_{n\to\infty}\,\frac{1}{n}\sum_{k=1}^n\,f\left(\frac{k}{n}\right)=\int_0^1f(x)\,dx. \end{equation}$

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Lesson Parent:

Integration IV: Definite Integrals As Riemann Sums

Problem:

Prove that if $f(x)$ is integrable between $0$ and $1$ , then
$\begin{equation} \lim_{n\to\infty}\,\frac{1}{n}\sum_{k=1}^n\,f\left(\frac{k}{n}\right)=\int_0^1f(x)\,dx. \end{equation}$

Solution:

Because we are explicitly told that $f(x)$ is Riemann integrable between $0$ and $1$ , then we know that for every <Sign in to see all the formulas> there exists a <Sign in to see all the formulas> such that for all partitions where <Sign in to see all the formulas> then

<Sign in to see all the formulas>

Now let's pick a $n$ such that <Sign in to see all the formulas>. Since the above is true for all partitions it means it is true for the partition <Sign in to see all the formulas>. Since each <Sign in to see all the formulas>, then <Sign in to see all the formulas>. If we choose <Sign in to see all the formulas>, then we have the Riemann sum

<Sign in to see all the formulas>

Now let's re-write the logical statement above but for only this partition. For every <Sign in to see all the formulas> there exists a <Sign in to see all the formulas> such that if <Sign in to see all the formulas> then

<Sign in to see all the formulas>

Because $n$ and <Sign in to see all the formulas> are greater than zero we can say that if <Sign in to see all the formulas>. Furthermore, let's define <Sign in to see all the formulas>. We can now say that for every <Sign in to see all the formulas> there exists a <Sign in to see all the formulas> such that if <Sign in to see all the formulas> then

<Sign in to see all the formulas>

By definition this means that

Prove that if f(x)f(x) is integrable between 00 and 11, then limn→∞1nn∑k=1f(kn)=∫10f(x)dx.\begin{equation} \lim_{n\to\infty}\,\frac{1}{n}\sum_{k=1}^n\,f\left(\frac{k}{n}\right)=\int_0^1f(x)\,dx. \end{equation}

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SAMPLE SOLUTION

<Sign in to see all the formulas>

<Sign in to see all the formulas>

<Sign in to see all the formulas>

<Sign in to see all the formulas>

<Sign in to see all the formulas>

GAIN AN ADVANTAGE

Prove that if $f(x)$ is integrable between $0$ and $1$ , then $\begin{equation} \lim_{n\to\infty}\,\frac{1}{n}\sum_{k=1}^n\,f\left(\frac{k}{n}\right)=\int_0^1f(x)\,dx. \end{equation}$