The definite integral is the key tool in calculus for defining and calculating quantities important to mathematics and science, such as areas, volumes, lengths of curved paths, probabilities, and the weights of various objects, just to mention a few.
The idea behind the integral is that we can effectively compute such quantities by breaking them into small pieces and then summing the contributions from each piece.
Riemann’s definition of integral
Figure 1 show that the area under a curve can be approximated using rectangular strips of different size as the sum (also called upper sum):
By varing the number of strips the approximation of the integral caan be improved. In the example given in the Figure 1a there are two strips therefore the area is given by
In the partition in Figure 1b the interval is and the new sum is given by
We can also use the lower sums to approximate the function as shown in Figure 2.
It gives us the approximate area of
Therefore the correct area is between the two summations
and if we increase the number of rectangles we can define the area as a limit of the value of the two summations. This limit is also very close to the one that we got using rectangles defined using the Middle Point Rule (Figure 3). In this case, the are is given by
that is only 0.5 % different from the analytical value (83.33) of the integral.
Therefore, the definition of the Riemann integral require a passage to the limit of the infinitesimal partition of the area underneath the interval limits. Given the interval the size of the interval is given by the number of segments , the Riemann integral is defined as
Rigorously the Riemann integral is defined as follows
Partition. Let be real number. A partition P of the interval is a finite subset of real numbers such that . We write .
Norm of a Partition. We define the norm of partition P, written, to be largest of all the subintervall widths. If is a small number then all the subintervalls in the partitionP have a small width.
The Riemann Integral
The be a function definied on a closed interval . The number is the definite integral of over and that S is the limit of the Riemann sum:
if the following condition is satisfied:
Given any number there is a corresponding number such that for every partition of with and any choice of in , we have
Definite integral properties
Here a list of some important properties of the definite integral:
Fundamental Theorem of Calculus
The definite integral is defined as
The function F(x) gives the area under the graph of from a to x when is nonnegative and .
If we denote the area between and as then
Theorem: If is continuous function on , then is continuous on and differentiable on and its derivative is :
Consider two functions and with c a constant. Then
Thus indefinite integrals are denoted as
Definition of Integral as antiderivative
Thus in the limit that , we obtain
Therefore, the function of is the inverse of the derivative (or antiderivative) of
By examining the last expression it is easy to find that only the values for , and the one for n=N, do not cancel out giving the demonstration of the following theorem.
Theorem: If is continuous at every point in and is any antiderivative of on , then
Some example indefinite integrals
or using the substitution and
Integral by Parts Method
The derivative of a product of two functions , can rearrange be rearranged as . By taking the integral of all the terms, we can write
or by introducing the two new functions and , as
 Maurice D. Weir, Joel Hass, George B. Thomas. Thomas’s Calculus. 12thEdition, Pearson.