The Fundamental Theorem of Integral Calculus

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Lesson Summary: 
The Fundamental Theorem of Integral Calculus is the main technique we use to determine the value of the definite integral $\int_a^b\,f(x)dx$. The theorem states that \[ \int_a^b\,f(x)dx=F(b)-F(a) \] where $F(x)$ is the antiderivative of $f(x)$.
Lesson Inputs: 
Antiderivatives
Integration III: Definite Integrals As Upper and Lower Sums
Integration IV: Definite Integrals As Riemann Sums
Lesson: 

The Fundamental Theorem of Integral Calculus is the primary technique we use to find the solution of integrals. Its importance cannot be overstated. Furthermore, it provides an elegant relationship between differentiation and integration. So let's get straight to it.

The Fundamental Theorem of Integral Calculus
If $f(x)$ is integrable on $[a,b]$ and $F(x)$ is an antiderivative of $f(x)$ on $[a,b]$, then
\[
\int_a^b\,f(x)dx=F(b)-F(a).
\]

A general proof can be found here and a proof for continuous functions can be found here.

There are times when we know that a function is integrable but not know what the number actually is. For example, we know that continuous functions are integrable. However, proving this large class of functions is integrable does not tell us what the number $\int_a^b\,f(x)dx$ is for a particular continuous function $f(x)$. We could make a guess as to what the number is and then for each guess use one of our three equivalent definitions of a definite integral to logically prove (or disprove) the guess. But let's be clear on this point. No one in their right mind wants to do this. It's too hard. This is where the Fundamental Theorem of Integral Calculus comes in. Once we know that a function is integrable it gives a quick way to determine what the actual number $\int_a^b\,f(x)dx$ is by finding the antiderivative of $f(x)$. Yes, this is very cool. The Fundamental Theorem of Integral Calculus is now your best friend.

Let's end the discussion by using the Fundamental Theorem of Integral Calculus to find EVERY definite integral of EVERY polynomial! The full proof can be found here. Basically we have the polynomial
\[
P_n(x)=c_nx^n+c_{n-1}x^{n-1}+\cdots +c_0
\]
and we want to find
\[
\int_a^b\,P_n(x)dx.
\]
We know the antiderivative of $P_n(x)$ is
\[
F(x)=\frac{c_n}{n+1}x^{n+1}+\frac{c_{n-1}}{n}x^n+\cdots +c_0x+C.
\]
Therefore,
\[
\int_a^b\,P_n(x)=\frac{c_n}{n+1}\left\{b^{n+1}-a^{n+1}\right\}+\frac{c_{n-1}}{n}\left\{b^n-a^n\right\}+\cdots+c_0\left\{b-a\right\}.
\]