Question

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

Answer 1

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.