Prove the Fundamental Theorem of Integral Calculus for continuous functions.

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Lesson Parent: 
The Fundamental Theorem of Integral Calculus
Problem: 

Prove the Fundamental Theorem of Integral Calculus for continuous functions which states:
if $f(x)$ is continuous on $[a,b]$ and $F(x)$ is an antiderivative of $f(x)$, then
\[
\int_a^b\,f(x)dx=F(b)-F(a).
\]

Answer: 

It is true that $F(b)-F(a)=\int_a^b\,f(x)dx$ where $F(x)$ is an antiderivative of $f(x)$.

SAMPLE SOLUTION

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Solution: 

The Fundamental Theorem of Integral Calculus for continuous functions states that if $f(x)$ is continuous on $[a,b]$ and $F(x)$ is an antiderivative of $f(x)$, then

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Let's define

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We know from this problem that $J(x)$ is an antiderivative of $f(x)$.

We know that antiderivatives are not unique. Let's say that we have another anitderivative $F(x)$. We know from this problem that <Sign in to see all the formulas>. So now let's look at <Sign in to see all the formulas>. Recall that

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Therefore, <Sign in to see all the formulas>.

Now let's look at <Sign in to see all the formulas>. Recall that

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is the integral that we are wanting to find. Plugging this in for $J(b)$ and $F(a)$ for $C$ we have

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The proof is now complete.

Before we leave this problem we want to address something that new students to Calculus may be wondering. If

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is what we are wanting to find then why not just find $J(x)$ by taking the antiderivative of $f(x)$ and find $J(b)$. But remember that $J(x)$ has a constant $C$ that you don't actually know. This means that there is an infinite number of antiderivatives that all differ from one another by a constant. Because of this $J(b)$ is usually not the real answer. It typically differs from the real answer by a constant we need to find. That constant turns out to be $-J(a)$. Since $F(x)$ and $J(x)$ are related by <Sign in to see all the formulas> then

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This means that

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is not just $J(b)$, but is <Sign in to see all the formulas>. Furthermore, it means that we can take ANY antiderivative of $f(x)$, evaluate it at $b$ and $a$, and then do the subtraction to get the integral.