Prove that \begin{equation} \int_a^a\,f(x)dx=0. \end{equation}

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Lesson Parent: 
Integration III: Definite Integrals As Upper and Lower Sums
Problem: 

Prove that
\begin{equation}
\int_a^a\,f(x)dx=0.
\end{equation}

Answer: 

It is true that \begin{equation} \int_a^a\,f(x)dx=0. \end{equation}

SAMPLE SOLUTION

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Solution: 

From our definition of a definite integral we know that

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where $m$ is the minimum of the function on $[a,b]$ and $M$ is the maximum value of the function on $[a,b]$. Furthermore, we know that all partitions $P$ will satisfy

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If we set $b=a$ then we know that <Sign in to see all the formulas> and <Sign in to see all the formulas> and we have

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Therefore,

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Before we end the proof we want to make one final comment. Our definition requires us to find a number for <Sign in to see all the formulas> such that

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for all possible partitions. However, in this case there are no partitions because there are no points between $a$ and$b$ (because $a=b$).