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Prove that $\int_0^1f(x)\,dx=0$ where $\begin{equation} f(x)=\left\{\begin{array}{c}1,\,\mbox{if }x=\frac{1}{2}\\ 0,\,\mbox{if }x\neq\frac{1}{2}\end{array}\right. \end{equation}$ using Riemann sums.

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Lesson Parent:

Integration IV: Definite Integrals As Riemann Sums

Problem:

Prove that $\int_0^1f(x)\,dx=0$ where
$\begin{equation} f(x)=\left\{\begin{array}{c}1,\,\mbox{if }x=\frac{1}{2}\\ 0,\,\mbox{if }x\neq\frac{1}{2}\end{array}\right. \end{equation}$
using Riemann sums.

Answer:

It is true that $\int_0^1f(x)\,dx=0$ where $\begin{equation} f(x)=\left\{\begin{array}{c}1,\,\mbox{if }x=\frac{1}{2}\\ 0,\,\mbox{if }x\neq\frac{1}{2}\end{array}\right.\end{equation}$

Solution:

To prove this we must show 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>

First, let's construct every possible $R(P)$ . For most useful functions this is humanly impossible, but for this simple function we can actually do it. We can do this by examining two cases. The first case is when no <Sign in to see all the formulas> is equal to $1/2$ . In this case all <Sign in to see all the formulas>'s are equal to zero and <Sign in to see all the formulas>. Showing that <Sign in to see all the formulas> is easy in this case because <Sign in to see all the formulas>.

The second case is when there is an <Sign in to see all the formulas> that is equal to $1/2$ . In this case <Sign in to see all the formulas> because all other points are zero. To be explicit we have

<Sign in to see all the formulas>

To do the proof we pick an arbitrary <Sign in to see all the formulas> which we simply denote as <Sign in to see all the formulas>, choose <Sign in to see all the formulas>. Now assume that we have a partition such that <Sign in to see all the formulas>. Now by the definition of $||P||$ , <Sign in to see all the formulas>. Therefore, <Sign in to see all the formulas>. Finally, since <Sign in to see all the formulas>, <Sign in to see all the formulas>.

So we have found every possible $R(P)$ and shown that for every <Sign in to see all the formulas> there exists a <Sign in to see all the formulas> (<Sign in to see all the formulas> in this problem) such that if <Sign in to see all the formulas> then <Sign in to see all the formulas>. Therefore,

Prove that ∫10f(x)dx=0\int_0^1f(x)\,dx=0 where f(x)={1,if x=120,if x≠12\begin{equation} f(x)=\left\{\begin{array}{c}1,\,\mbox{if }x=\frac{1}{2}\\ 0,\,\mbox{if }x\neq\frac{1}{2}\end{array}\right. \end{equation} using Riemann sums.

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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>

GAIN AN ADVANTAGE

Prove that $\int_0^1f(x)\,dx=0$ where $\begin{equation} f(x)=\left\{\begin{array}{c}1,\,\mbox{if }x=\frac{1}{2}\\ 0,\,\mbox{if }x\neq\frac{1}{2}\end{array}\right. \end{equation}$ using Riemann sums.