Section 5.4 The Fundamental Theorem of Calculus
We are now getting into the truly amazing core of calculus. We mentioned a number of times that “differentiation and integration are inverse processes”. But what does this mean precisely? This is the meat of the Fundamental Theorem of Calculus.
In fact, in previous sections, we used pretty much the same symbol, \(\displaystyle \int\text{,}\) to first denote the indefinite integral of a function (which is its most general antiderivative - this was the symbol without limits of integration), and second the definite integral of a function (which is the limit of a Riemann sum - this was the symbol with limits of integration). A priori, those are two very different concepts. Why do we use the same symbol for both? This is because they are very intimately related: it turns out that we can evaluate definite integrals in terms of arbitrary antiderivatives. Again, this is encapsulated into the all-important Fundamental Theorem of Calculus. So let's dive right into it!
Objectives
You should be able to:
- State the Fundamental Theorem of Calculus (FTC).
- Sketch the main lines of the proof of the FTC.
- Explain in which sense the FTC is saying that differentiation and integration are inverse processes.
- Use the FTC Part 1, in conjunction with the chain rule and properties of definite integrals, to evaluate the derivatives of functions presented as integrals.
- Use the FTC Part 2 to evaluate integrals in terms of antiderivatives.
Subsection 5.4.1 Instructional video
Subsection 5.4.2 Key concepts
Concept 5.4.1. The Fundamental Theorem of Calculus (FTC).
Let \(f(x)\) be a continuous function on \([a,b]\text{.}\) Then:
- If \(\displaystyle g(x) = \int_a^x f(t)\ dt\text{,}\) for \(a \leq x \leq b\text{,}\) then \(g'(x) = f(x)\text{;}\)
- \(\displaystyle \int_a^b f(x)\ dx = F(b) - F(a)\text{,}\) where \(F\) is an arbitrary antiderivative of \(f\text{.}\)
The FTC gives a precise meaning to the statement that integration and differentiation are inverse processes.
Concept 5.4.2. Some uses of the FTC.
- The FTC part 1 can be used to evaluate derivatives of functions that are given in integral form. Note that in part 1 above, \(g(x)\) is a function of \(x\text{,}\) not of the dummy variable \(t\) that is integrated over.
- The FTC part 2 can be used to evaluate definite integrals, by first finding an antiderivative of the integrand.