Advanced calculus concepts and applications

Calculus and Definitions of Its Concepts

Indefinite integration

Indefinite integration is the act of reversing any process of differentiation. It is the process of obtaining a function from its derivative. It is also called anti-derivative of f. A function F. is an anti-derivative of f on an interval I, if F'(x) = f (x) for all x in I. A function of F (x) for which F'(x)=f (x), this means that for every x domain of f is said to be an anti-derivative of f (x)

The anti-derivative of a derivative is the original function plus a constant. In most cases indefinite integral is denoted by ? symbol which is called the integral sign, and f (x) is referred to as the integrand. In most cases in indefinite integration the constant C. is always zero this means that any constant can be added to it and the corresponding function bear the same integral.

An indefinite integral is in the form:

If the bounds are not specified, then the integral is indefinite, and it no longer corresponds to a particular numeric value. There is a simple geometric interpretation for the fact that any two anti-derivatives of the same continuous f differ by at most a constant. When we say that F. And G. are both anti-derivatives of f we mean that F'(x) = G'(x) therefore the slope of the curve y = f (x) is the same as that of y = G (x) in other words the graph of G (x) is a vertical translation of the graph F (x) (Bradley et al.,.2000)

Indefinite integral differs from definite integral in that the indefinite integral exists it usually exists as a real value, while the values vary according to the constant.

Definite integration

The formal definition of a definite integral is stated in terms of the limit of a Riemann sums. We will introduce the definite integral defined in terms of area. Whereas the indefinite integration analyzes situations involving the reversal of the rate of change, definite integration involves the definition of the limit of a sum and then it is later computed using the anti-differentiation.

Consider the area A of the region under the curve y = f (x) above an interval a ? x ? b, where f (x) ? 0 and f is continuous. If the region was a square, trapezoid or triangular, we could find its area using well-known formulas but if the bounding curve were y = x2 or y = ex-when such a situation occurs we apply the general mathematics policy 'When faced with something unknown, relate it to something known'.

In this particular case we may not know the area under the given curve, but we do know how to find the area of a rectangle, thus we proceed by sub-dividing the region into a number of rectangular regions and then approximate the area under the curve y = f (x) by adding the areas of approximating rectangles (Decker 1996)

The region of integration is only well defined for a definite integral, which is one for which the bounds are specified. To clearly define a definite integral we may begin with the function f (x) which continues on closed intervals (c & d) we assume that these intervals are subdivided into sub-sections which we can name them (m). These subsections can be assumed to be of equal lengths which are ?x. Therefore the product of each of these product value and the corresponding sub-interval value (m) are added to determine the sum, these sum is what we are referring to as Riemann sum. The value may be a negative, positive or even a zero. In the event that the numbers of the subsections are repeatedly increased, the net effect would be that the length of each would be getting smaller therefore we can say that if the number of the subinterval is increased without bond then the length would approach zero. In simple terms the Riemann sum is used to define the definite integral of a function. We begin by considering a simple case, whereby the definition of the Riemann integral of a continuous function f over a rectangle R. In this case rather than having a one-variable case, we can overcome the tendency of connecting integration too strongly with anti-differentiation. (Buck 2003)

Formal definition

It is the definition of a function by use of graphs to define the limits; it uses Greek letters epsilon (?) and Delta (?). Epsilon always represents any distance on the limiting side and delta represents the distance on the x- axis. The limits of a given function clearly explain how that given function behaves when it nears the x value.

Consider the following functions g (x) and f (x), this functions are as a result of definition of real numbers. The following relationship exist x ?

This relationship exist only when there is a positive constant C. such that for all sufficiently large values of x, f (x) is at most C. multiplied by g (x) in absolute value. That is, f (x) = O (g (x)) if and only if there exists a positive real number C. And a real number x0 such that

In general the growth rate are of much interest in that the variable x which goes to infinity is often left unstated, and one writes more simply that f (x) = O (g (x)).

Additional explanation indicates that it doesn't matter how close a function can be to a limit, it is always necessary to find the corresponding x value which is closer to the given value and using the new notations of epsilon (?) and delta (?), we make f (x) within ? Of L, the limit, and later determine x within ? Of C. (Bradley et al.,.2000)

Again, since this is tricky, let's resume our example from before: f (x) =x2at x=2. To start, let's say we want f (x) to be within .01 of the limit. We know by now that the limit should be 4, so we say: for ?= 0.1, there is some ? so that as long as, then

To show this, we can pick any delta (?) that is bigger than 0, so long as it works. For example, you might pick .000000001, because you are absolutely sure that if x is within .00000000000001 of 2, then f (x) will be within .01 of 4. This works for. But we can't just pick a specific value for, like .01, because we said in our definition "for every." This means that we need to be able to show an infinite number of s, one for each.