The Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus
Objectives
- Describe the meaning of the Mean Value Theorem for Integrals.
- State the meaning of the Fundamental Theorem of Calculus, Part 1.
- Use the Fundamental Theorem of Calculus, Part 1, to evaluate derivatives of integrals.
- State the meaning of the Fundamental Theorem of Calculus, Part 2.
- Use the Fundamental Theorem of Calculus, Part 2, to evaluate definite integrals.
- Explain the relationship between differentiation and integration.
Summary
We can find the exact value of a definite integral without taking the limit of a Riemann sum or using a familiar area formula by finding the antiderivative of the integrand, and hence applying the Fundamental Theorem of Calculus.
The Fundamental Theorem of Calculus says that if \(f\) is a continuous function on \([a,b]\) and \(F\) is an antiderivative of \(f\text{,}\) then
\begin{equation*} \int_a^b f(x) \, dx = F(b) - F(a)\text{.} \end{equation*}Hence, if we can find an antiderivative for the integrand \(f\text{,}\) evaluating the definite integral comes from simply computing the change in \(F\) on \([a,b]\text{.}\)
See the Desmos Demonstration on Evaluating Definite Integrals.
A slightly different perspective on the FTC allows us to restate it as the Total Change Theorem, which says that
\begin{equation*} \int_a^b f'(x) \, dx = f(b) - f(a)\text{,} \end{equation*}for any continuously differentiable function \(f\text{.}\) This means that the definite integral of the instantaneous rate of change of a function \(f\) on an interval \([a,b]\) is equal to the total change in the function \(f\) on \([a,b]\text{.}\)