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

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Lesson Parent:

Integration IV: Definite Integrals As Riemann Sums

Problem:

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

Answer:

It is true that $\int_0^2f(x)\,dx=-1$ .

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 the partition contains the point $x=1$ . In other words, <Sign in to see all the formulas>. This effectively breaks $R(P)$ into two sums:

Now let's do the proof for this case. Pick an arbitrary <Sign in to see all the formulas> and denote it by <Sign in to see all the formulas>. Chose <Sign in to see all the formulas>. Now in this case not only do we have that <Sign in to see all the formulas>, but we also have constrained our partition to have an interval where <Sign in to see all the formulas>. Now let's do some simple algebra utilizing our rules of inequalities:

We have now shown that for every <Sign in to see all the formulas> there is a <Sign in to see all the formulas> (<Sign in to see all the formulas>) such that for every partition where <Sign in to see all the formulas> then <Sign in to see all the formulas>. Therefore,

Prove that ∫20f(x)dx=−1\int_0^2f(x)\,dx=-1 where f(x)={1,if 0≤x<1−2,if 1≤x≤2\begin{equation} f(x)=\left\{\begin{array}{c}1,\,\mbox{if }0\leq x\lt 1\\ -2,\,\mbox{if }1\leq x\leq 2\end{array}\right. \end{equation} using Riemann sums.

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SAMPLE SOLUTION

GAIN AN ADVANTAGE

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