### 5.3: The Fundamental Theorem of Calculus

Hey guys, this is a brief update on the Fundamental Theorem of Calculus. We learned the first part of it in the previous section, which is:

If *f* is continuous on [*a,b*], then the function *g* defined by

is continuous on [*a,b*] and differentiable on *(a,b)* and *g'(x) = f(x)*.

The Fundamental Theorem of Calculus Part II is as follows:

where *F* is the antiderivative of *f* so that *F' = f*.

Let's try a problem now, shall we?

So substitute 0 for a and (pi/2) in for b.

so the area of the graph is pi^2 +5.

sophie, i believe you are next up.

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