# Fundamental Theorem of Calculus

“The Fundamental Theorem of Calculus” is a big name, and it sounds like it’s a very important theorem. It is. Let’s state it first, and then talk about why it’s so important.

Definition:  Let $f(x)$ be a continuous function on $[a, b]$, and let $F(x)$ be an antiderivative of f. Then:

Why is this so important? Well, it means that we can use antiderivatives to calculate a definite integral. Recall the limit definition of a definite integral:

It is definitely easier to take the antiderivative than to evaluate this limit of a sum. The fundamental theorem of calculus saves us a lot of work!

Example: Find the area under $f(x) = x^2$  from $latex x=0 \text{ to } x=1$.

This time, we can use antiderivatives. Note that $F(x) = \int x^2 dx = \frac{x^3}{3} + C$  .  Then, using the fundamental theorem of calculus, we know that

Note that the +Cs cancel out. While it doesn’t matter which antiderivative we pick, it DOES matter that we pick the same antiderivative for F(b) as we do for F(a). Therefore, we usually pick C=0 when evaluating definite integrals.

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