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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