The Fundamental Theorem of Calculus (Part 2)

FTC 2 relates a definite integral of a function to the net change in its antiderivative.

Fundamental Theorem of Calculus (Part 2):   If $f$ is continuous on $[a,b]$, and $F'(x)=f(x)$, then $$\int_a^b f(x)\, dx = F(b) - F(a).$$

This FTC 2 can be written in a way that clearly shows the derivative and antiderivative relationship, as
$$\int_a^b g'(x)\,dx=g(b)-g(a).$$

This gives us an incredibly powerful way to compute definite integrals:

  1. Find an antiderivative.
  2. Evaluate it at the limits of integration.
This computation is the most important use of FTC 2 in this course.

Example:  DO:  use the FTC to evaluate $\displaystyle\int_1^3 3x^2\,dx$.

The following video gives examples of using FTC 2 to evaluate definite integrals.



The following video explains FTC 2.