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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 $f(x)$ .

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

<Sign in to see all the formulas>

Let's define

<Sign in to see all the formulas>

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

<Sign in to see all the formulas>

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

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

<Sign in to see all the formulas>

is the integral that we are wanting to find. Plugging this in for

$J(b)$ and

$F(a)$ for

$C$ we have

<Sign in to see all the formulas>

The proof is now complete.

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

<Sign in to see all the formulas>

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

<Sign in to see all the formulas>

This means that

<Sign in to see all the formulas>

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.