Integration vs Summation
In above high school mathematics, integration and summation are often found in mathematical operations. They are seemingly used as different tools and in different situations, but they share a very close relationship.
More about Summation
Summation is the operation of adding a sequence of numbers and the operation is often denoted by the Greek letter of capital sigma Σ. It is used to abbreviate the summation and equal to the sum/total of the sequence. They are often used to represent the series, which essentially are infinite sequences summed up. They can also be used to indicate the sum of vectors, matrices, or polynomials.
The summation is usually done for a range of values that can be represented by a general term, such as a series which has a common term. The starting point and the end point of the summation are known as the lower bound and upper bound of the summation, respectively.
For example, the sum of the sequence a1, a2, a3, a4, …, an is a1 + a2 + a3 + … + an which can be easily represented using the summation notation as ∑ni=1 ai; i is called the index of summation.
Many variations are used for the summation based on the application. In some cases, the upper bound and lower bound can be given as an interval or a range, such as ∑1≤i≤100 ai and ∑i∈[1,100] ai. Or it can be given as a set of numbers like ∑i∈P ai , where P is a defined set.
In some cases, two or more sigma signs can be used, but they can be generalized as follows; ∑j ∑k ajk = ∑j,k ajk.
Also, the summation follows many algebraic rules. Since the embedded operation is the addition, many of the common rules of algebra can be applied to the sums itself and for the individual terms depicted by the summation.
More about Integration
The integration is defined as the reverse process of differentiation. But in its geometric view it can also be considered as the area enclosed by the curve of the function and the axis. Therefore, calculation of the area gives the value of a definite integral as shown in the diagram.
Image Source: http://en.wikipedia.org/wiki/File:Riemann_sum_convergence.png
The value of the definite integral is actually the sum of the small strips inside the curve and the axis. The area of each strip is the height×width at the point on the axis considered. Width is a value we can choose, say ∆x. And height is approximately the value of the function at the considered point, say f(xi). From the diagram, it is evident that the smaller the strips are better the strips fit inside the bounded area, hence better approximation of the value.
So, in general the definite integral I, between the points a and b (i.e in the interval [a,b] where a<b), can be given as I ≅ f(x1)∆x + f(x2)∆x + ⋯ + f(xn)∆x , where n is the number of strips (n=(b-a)/∆x). This summation of the area can be easily represented using the summation notation as I ≅ ∑ni=1 f(xi)∆x. Since the approximation is better when ∆x is smaller, we can compute the value when ∆x→0. Therefore, it is reasonable to say I = lim∆x→0 ∑ni=1 f(xi)∆x.
As a generalization from the above concept, we can choose the ∆x based on the considered interval indexed by i (choosing the width of the area based on the position). Then we get
I=lim∆x→0 ∑ni=1 f(xi) ∆xi = a∫b f(x)dx
This is known as the Reimann Integral of the function f(x) in the interval [a,b]. In this case a and b are known as the upper bound and lower bound of the integral. Reimann integral is a basic form of all integration methods.
In essence, integration is the summation of the area when the width of the rectangle is infinitesimal.
What is the difference between Integration and Summation?
• Summation is adding up of a sequence of numbers. Usually, the summation is given in this form ∑ni=1 ai when the terms in the sequence have a pattern and can be expressed using a general term.
• Integration is basically the area bounded by the curve of the function, the axis and upper and lower limits. This area can be given as the sum of much smaller areas included in the bounded area.
• Summation involves the discrete values with the upper and lower bounds, whereas the integration involves continuous values.
• Integration can be interpreted as a special form of summation.
• In numerical computation methods, integration is always performed as a summation.