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

If we can find the antiderivative function $F \left(x\right)$ of the integrand $f \left(x\right)$, then the definite integral ${\int}_{a}^{b} f \left(x\right) \mathrm{dx}$ can be determined by $F \left(b\right) - F \left(a\right)$ provided that $f \left(x\right)$ is continuous.
We are usually given continuous functions, but if you want to be rigorous in your solutions, you should state that $f \left(x\right)$ is continuous and why.
For most students, the proof does give any intuition of why this works or is true. But let's look at $s \left(t\right) = {\int}_{a}^{b} v \left(t\right) \mathrm{dt}$. We know that integrating the velocity function gives us a position function. So taking $s \left(b\right) - s \left(a\right)$ results in a displacement.