The Fundamental Theorem

Section 8.3 The Fundamental Theorem

Subsection 8.3.1 Integrals as Functions

As mentioned in Section 8.2, I never actually want to use the definition to calculate integrals. With derivatives, I developed a series of rules for various types and combinations of functions. That doesn’t work as easily or cleanly for integrals. However, there is a very powerful theorem that gives an approach to solving integrals.

To state the theorem, I need to consider a strange new function. For \(g(t)\) continuous on \([a,b]\) and \(x \in [a,b)\text{,}\) define a new function \(f\text{.}\)

\begin{equation*} f(x) := \int_a^x g(t) dt \end{equation*}

Let me give some interpretation: \(f(x)\) is the area under the function \(g(t)\) along the \(t\) axis between a fixed point \(t=a\) and a varying endpoint \(x\text{.}\) Be careful to keep the variables straight! The variable \(t\) is only inside the integral and the variable \(x\) is only outside. \(x\) is the endpoint of the interval and \(t\) is the variable that goes between \(a\) and \(x\) on the interval.

Though very strange, this definition is a rich source of new and interesting functions in mathematics. Here are some examples of functions that are defined as integrals of other functions.

Example 8.3.1.

\begin{align*} \text{Fresnel Function} \amp \amp S(x) \amp \defeq \int_0^x \sin \left(\frac{\pi t^2}{2} \right) dt\\ \text{Logarithmic Integral} \amp \amp li(x) \amp \defeq \int_0^x \frac{1}{\ln t} dt\\ \text{Sine Integral} \amp \amp Si(x) \amp \defeq \int_0^x \frac{\sin t}{t} dt \end{align*}

Subsection 8.3.2 The Fundamental Theorem

Now that I’ve defined the function \(f(x)\text{,}\) I can look at its derivative. Figure 8.3.2 gives a useful visualization. In the figure, I’ve labelled a point \(x\) and a small increase \(x+h\text{.}\)

Figure 8.3.2. The Area Function \(f(x)\)

I use the limit definition of the derivative to calculate this derivative.

\begin{equation*} \frac{d}{dx} f(x) = \lim_{h \rightarrow 0} \frac{f(x+h) - f(x)}{h} \end{equation*}

Since \(f\) measures area under \(g(t)\text{,}\) the difference \(f(x+h) - f(x)\) is just the area of the very thin rectangle in Figure 8.3.2. When \(h\) is small, this area is roughly the height \(g(x)\) times the width \(h\text{.}\)

\begin{equation*} \frac{d}{dx} f(x) = \lim_{h \rightarrow 0} \frac{g(x) h}{h} = g(x) \end{equation*}

All this work can be summarized in the following statement, which is called the Fundamental Theorem of Calculus.

\begin{equation*} \frac{d}{dx} f(x) = \frac{d}{dx} \int_a^x g(t) dt = g(x) \end{equation*}

The important implication of the fundamental theorem is this: integral and derivatives are opposite operations. In the previous statement, I started with a function \(g\text{,}\) took an integral to get \(f\) and then took a derivative to get back to \(g\text{.}\) Derivatives and integrals are reverse processes.

There are several versions of the fundamental theorem, but they all capture this basic idea of reserve processes. The derivative can be inside the integral.

\begin{equation*} \int_a^b f^\prime(t) dt = f(b) - f(a) \end{equation*}

This version of the fundamental theorem gives a wonderful way to calculate definite integrals. Let \(F\) be any function such that \(F^\prime(x) = f(x)\text{.}\) Such a function \(F\) is called an antiderivative for \(f\text{.}\) I use antiderivatives to evaluate integrals.

\begin{equation*} \int_a^b f(t) dt = F(b) - F(a) \end{equation*}

Subsection 8.3.3 The Indefinite Integral

To calculate antiderivatives, I need to do deriavitves backwards. Fortunately, this is much easier than using the definition of the integral; it is the main strategy for integrals. Unfortunately, it is still quite difficult for many functions.

It’s useful to have a notation for antiderivatives. Since integrals are, in some way, the inverse operation, it seems natural to use an integral symbol to indicate an antiderivative. I drop the bounds that I used for definition integrals.

\begin{equation*} \text{ Any antiderivative of f is written } \int f(x) dx \end{equation*}

This is called an indefinite integral. Note that the notation means all anti-derivatives, since there may be more than one.

Example 8.3.3.

In these examples, I’m using what I know about derivatives to do the operation backwards. The sixth example is just the power rule done backwards. To check any of these is not difficult: just differentiate the right side of the equation. Since they are anti-derivatives, the derivatives of the right side should give the original function on the left (inside the integral sign). The \(+ c\) is an arbitrary constant which must be added. It shows up because it would disappear in differentiation.

\begin{align*} \int e^x dx \amp = e^x + c\\ \int \sin x dx \amp = -\cos x + c\\ \int \cos x dx \amp = \sin x + c\\ \int a^x dx \amp = \frac{a^x}{\ln a} + c\\ \int \frac{1}{1+x^2} dx \amp = \arctan x + c\\ \int x^n dx \amp = \frac{x^{n+1}}{n+1} + c\\ \int \frac{1}{x} \amp = \ln |x|\\ \int \frac{1}{\sqrt{1-x^2}} dx \amp = \arcsin x + c\\ \int \sec^2 x dx \amp = \tan x + c\\ \int \csc^2 x dx \amp = \cot x + c \end{align*}

Subsection 8.3.4 Calculating Definite Integrals

I can use the fundamental theorem to calculate definite integrals as well. There is some special notation that is useful here: when I write \(\left. F(x) \right|_a^b\text{,}\) I mean to evaluate the function \(F\) at \(a\) and \(b\) and subtract the two evaluations.

\begin{equation*} F(x) \Big|_a^b = F(b) - F(a) \end{equation*}

Then the integral is the antiderivative evaluated with this notation.

\begin{equation*} \int_a^b f(x) dx = F(x) \Big|_a^b = F(b) - F(a) \end{equation*}

Example 8.3.4.

The antiderivative of \(x^2\) is \(\frac{x^3}{3}\text{.}\) This antiderivative lets me calculate the following integral.

\begin{equation*} \int_2^4 x^2 dx = \left. \frac{x^3}{3} \right|_2^4 = \frac{4^3}{3} - \frac{2^3}{3} = \frac{64 - 8}{3} = \frac{56}{3} \end{equation*}

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