MathWithNaziaa : Integration

Integration :

Integration is the reverse of differentiation.

Definition : In mathematics, technique of finding a function g(x) the derivative of which, Dg(x), is equal to a given function f(x). This is indicated by the integral sign “∫,” as in ∫f(x), usually called the indefinite integral of the function. The symbol dx represents an infinitesimal displacement along x; thus ∫f(x)dx is the summation of the product of f(x) and dx. The definite integral, written 

with a and b called the limits of integration, is equal to g(b) − g(a), where Dg(x) = f(x).

If y = 2x + 3, dy/dx = 2
If y = 2x + 5, dy/dx = 2
If y = 2x, dy/dx = 2

So the integral of 2 can be 2x + 3, 2x + 5, 2x, etc.
For this reason, when we integrate, we have to add a constant. So the integral of 2 is 2x + c, where c is a constant.
A "S" shaped symbol is used to mean the integral of, and dx is written at the end of the terms to be integrated, meaning "with respect to x". This is the same "dx" that appears in dy/dx .

To integrate a term, increase its power by 1 and divide by this figure. In other words:

When you have to integrate a polynomial with more than 1 term, integrate each term. So:

 

Basic integration formulas

 

The fundamental use of integration is as a continuous version of summing. But, paradoxically, often integrals are computed by viewing integration as essentially an inverse operation to differentiation. (That fact is the so-called Fundamental Theorem of Calculus.)

 Definite Integral :

Definition and Notation

The

definite integral

generalizes the concept of the area under a curve. We lift the requirements that f(x)

be continuous and nonnegative, and define the definite integral as follows.

Definition:

Definite Integral

If f(x) is a function defined on an interval [a,b], the definite integral of f from a to b

is given by

provided the limit exists. If this limit exists, the function f(x) is said to be integrable on [a,b], or is an  integrable function.

The integral symbol in the previous definition should look familiar. We have seen similar notation in the chapter on Applications of Derivatives, where we used the

indefinite integral symbol (without the a and b above and below) to represent an antiderivative. Although the notation for indefinite integrals may look similar to the notation for a

definite integral, they are not the same. A definite integral is a number. An indefinite integral

is a family of functions. Later in this chapter we examine how these concepts are related. However, close attention should always be paid to notation so we know whether we’re working with a definite integral or an indefinite integral.

Integral notation goes back to the late seventeenth century and is one of the contributions of Gottfried Wilhelm Leibniz, who is often considered to be the codiscoverer of calculus, along with Isaac Newton. The integration symbol ∫ is an elongated S, suggesting sigma or summation. On a definite integral, above and below the summation symbol are the boundaries of the interval, [a,b]. The numbers a and b are x-values and are called the limits of integration; specifically, a is the lower limit and b is the upper limit. To clarify, we are using the word limit in two different ways in the context of the definite integral. First, we talk about the limitof a sum as n→∞.

Second, the boundaries of the region are called the limits of integration.

We call the function f(x) the integrand, and the dx indicates that f(x) is a function

with respect to x, called the variable of integration. Note that, like the index in a sum, the

variable of integration is a dummy variable, and has no impact on the computation of the integral. We could use any variable we like as the variable of integration:

 

Previously, we discussed the fact that if f(x) is continuous on [a,b], then the

Limit

 

exists and is unique. This leads to the following theorem, which we state without proof.

Continuous Functions Are Integrable

If f(x)

is continuous on [a,b], then f is integrable on [a,b].

Functions that are not continuous on [a,b]

may still be integrable, depending on the nature of the discontinuities. For example, functions with a finite number of jump discontinuities or removable discontinuities on a closed interval are integrable.

It is also worth noting here that we have retained the use of a regular partition in the Riemann sums. This restriction is not strictly necessary. Any partition can be used to form a Riemann sum. However, if a nonregular partition is used to define the definite integral, it is not sufficient to take the limit as the number of subintervals goes to infinity. Instead, we must take the limit as the width of the largest subinterval goes to zero. This introduces a little more complex notation in our limits and makes the calculations more difficult without really gaining much additional insight, so we stick with regular partitions for the Riemann sums.