How do you use the Fundamental Theorem of Calculus to evaluate an integral?

1 Answer
Amory W.
Aug 20, 2014

If we can find the antiderivative function F(x) of the integrand f(x), then the definite integral int_a^b f(x)dx can be determined by F(b)-F(a) provided that f(x) is continuous.

We are usually given continuous functions, but if you want to be rigorous in your solutions, you should state that f(x) is continuous and why.

FTC part 2 is a very powerful statement. Recall in the previous chapters, the definite integral was calculated from areas under the curve using Riemann sums. FTC part 2 just throws that all away. We just have to find the antiderivative and evaluate at the bounds! This is a lot less work.

For most students, the proof does give any intuition of why this works or is true. But let's look at s(t)=int_a^b v(t)dt. We know that integrating the velocity function gives us a position function. So taking s(b)-s(a) results in a displacement.

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