Notation & Concepts

We start off with the definition of the anti-derivative:

Given a function $f(x)$ an anti-derivative of $f(x)$ is a function, say $F(x)$, whose derivative is $f(x)$ such that

The most general anti-derivative of $f(x)$ is called an indefinite integral,

where $c$ is the constant of integration. Here, $f(x)$ is called the integrand. Indefinite integrals therefore are defined up to an additive constant: if $F(x)$ is an antiderivative for $f(x)$ so is $F(x)+c$.

Properties of the indefinite integral

I. $\int k f(x) d x=k \int f(x) d x$ where $k$ is any number;

II. $\int f(x) \pm g(x) d x=\int f(x) d x \pm \int g(x) d x$.

Note that in integration, we do not have rules similar to the product and quotient rules in differentiation. The technique of integration followed in such cases depends on the particular integrand in question.

There are three ways of thinking about integrals:

1. Geometrically, as the area under a graph;
2. Analytically, as the limit of a sum;
3. Deriving from anti-derivatives given by the Fundamental Theorem of Calculus.

In what follows, we look at each case in more detail.

1. Geometric definition

Suppose $f(x)$ is defined for $a \leq x \leq b$. The notation

with $b>a$ denotes the area under the curve $y=f(x)$ and above the $x$-axis for $a \leq x \leq b$, as shown below.

If the graph falls below the $x$-axis, the area above the curve and below the $x$-axis counts negatively (i.e. $A_{2}$ in the plot below).

Therefore,

Note that, if $b<a$,

Integrals can sometimes be computed via simple geometric reasoning. Some examples are given below.

(i)

gives the area of a rectangle with sides 1 and $b-a$;

(ii)

gives half the area of a circle of radius 1 ;

(iii)

by symmetry since

2. Analytic definition

Try to approximate

as the area of a finite number $N$ of rectangles. Let $x_{1}, x_{2}, \cdots, x_{N-1}$ satisfy

and define $x_{0}=a$ and $x_{N}=b$. This is shown in the next figure from which we can see that,

the above sum is known as the Riemann sum.

Again, rectangles below the $x$-axis count negatively. As the intervals between successive points in $x$ get smaller and smaller, $N \rightarrow \infty$; one can define the integral

as a limit where the number of rectangles is taken to infinity.

3. Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus states that if $F^{\prime}(x)=f(x)$ and $f(x)$ is a continuous function on $[a, b]$ then,

We consider

where $a=x_{0}, b=x_{N}$, as above; then $F(b)-F(a)$ can be written as

We can approximate the term in red above by $F^{\prime}\left(x_{m-1}\right)$ which in turn is equal to $f\left(x_{m-1}\right)$ given that $F^{\prime}(x)=f(x)$. Therefore, Eq. (8.8) gives

Letting $N \rightarrow \infty$

To compute the integral above, we need to find the anti-derivative, $F(x)$ such that $F^{\prime}(x)=f(x)$. For instance, if we take the integral $y=f(x)=x^{2}$, we note that $F(x)=x^{3} / 3$ is an anti-derivative for $f(x)=x^{2}$ so that

Some common indefinite integrals are:

Definite & improper integrals

Computing definite integrals is straightforward once an anti-derivative integral is known. We have already carried out an example when discussing the Fundamental Theorem of Calculus above. For the integral

an anti-derivative is $F(x)=e^{x}$ so that

It is customary to use the bar notation rather than the anti-derivative $F$,

Here, $\left.g(x)\right|_{a} ^{b}$ is shorthand notation for $g(b)-g(a)$.

Earlier we interpreted

as the limit of a sum. Such definitions do not work for integrals of the form,

These can be defined as limits of definite integrals,

These are called improper integrals. Consider

since $\lim _{b \rightarrow \infty} e^{-b}=0$.