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The Fundamental Theorem of Calculus allows us to integrate a function between two points by finding the indefinite integral and evaluating it at the endpoints. The equation is \[ \int_{a}^{b}{f(x)~dx} = \left. F(x) \right|_{a}^{b} = F(b) - F(a) \] where \(F' = f\).
So what is this theorem saying? In essence, it says we only have to calculate the indefinite integral and plug in the endpoints, subtract them in the right order and the value we get is the area under a curve.

Okay, let's take a few minutes and watch a couple of videos explaining the proof of this theorem.

Notice in the equation above that, after we evaluate the integral, we use a vertical bar in order to 'hang' the limits of integration before we do the substitution. This notation is very common. You will also see some instructors use a right bracket instead of a vertical bar. We prefer the vertical bar unless we also use a left bracket to enclose the \(F(x)\) expression. So on this site, you will see either \(\displaystyle{ \left. F(x) \right|_{a}^{b} }\) or \(\displaystyle{ \left[ F(x) \right]_{a}^{b} }\).

Another way to write this is to explicitly write the variable that the limits of integration will be substituted into, like this \(\displaystyle{ \left. F(x) \right|_{x=a}^{x=b} }\). This is a very good way to do write it but is not very common. If your instructor allows it, we recommend it since, when you get to multi-variable calculus you will be required to write it this way. However, as usual check with your instructor to see what they require.

Before we go on, let's work some basic integrals using this theorem. After that, your next logical step is to learn about the Second Fundamental Theorem of Calculus.