A definite integral is simple called as an integral .It also has both upper and lower limits. If x is restricted to lie on the real line ,then the definite integral is known as a Riemann integral. However a general definite integral is taken in the complex plane, resulting in the contour integral
Here a,b and z in general being complex numbers and the path of integration from a to b which is known as a contour.
The term integral may also be refer to the notion of antiderivative, a function F whose derivative is the given function ƒ. In this case it is called as an indefinite integral. The first fundamental theorem of calculus which allows definite integrals to be computed in terms of indefinite integrals since if f is the indefinite integral for a continuous functionf(x) then
A definite integral of a function can be represented as the signed area of the region bounded by its graph.
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Types of definite integral
1 Reimann integral.
2 Lebesgue integral.
3 Other integral.
1 Reimann integral.
The Riemann integral is defined in terms of Riemann sums which is of functions with respect to tagged partitions of an interval . Let [a,b] be a closed interval of the real line therefore there is a a tagged partition of [a,b] is a finite sequence
Riemann sums converging as intervals halve, whether sampled at - right, - minimum, - maximum, or - left.
2 Lebesgue integral.
The Riemann integral is not defined for a wide range of functions and situations of importance in applicationsthrefore lebesgue integral formed. The Lebesgue integral, in particular, achieves great flexibility by directing attention to the weights in the weighted sum. The Lebesgue integral begins with a measure, μ . In this the simplest case the Lebesgue measure μ(A) which is an interval of A = [a,b] and its width is b − a so that the Lebesgue integral agrees with the proper Riemann integral .
3 Other integral.
The Riemann and Lebesgue integrals are the most widely used definitions of the integral but there are a number of others integral exist for example.
Properties of definite integral
The properties of definite integral are very useful in the calculation of the integral, as well as completing your understanding of the basic of integration.
In this suppose the f is integrable over the interval[ a; b ] and c £ R, then cf is integrable over the interval [ a; b ]
The Additive of Integrand
In this suppose the functions f and g are integrable over the interval [ a; b ] then f + g, is integrable over the interval [ a; b ]
This statement is also true for the difference of two functions.
Additivity of Limits.
If f is integrable over the interval [ a; b ], and let c £ [ a; b ]
Nonnegativity of the Integral.
The function f is integrable over [ a; b ] and that f(x) ≥ 0 over [ a; b ]
The Riemann integrable functions on a closed interval [a, b] forms a vector space which is under the operations of the pointwise addition and the multiplication by a scalar, and the operation of integration
is a linear functional .The collection of integrable functions is closed under taking linear combinations and secondly the integral of a linear combination is the linear combination of the integrals.
Inequalities for integrals
A number of general inequalities is hold for the Riemann-integrable functions which defined a closed and bounded interval [a, b] and can be generalized to other notions of integral .
Upper and lower bounds
An integrable function f on [a, b], is necessarily bounded on that interval. Thus there are real numbers m and M so that the m ≤ f (x) ≤ M for all x in [a, b]. Since the lower and upper sums of f over [a, b] are therefore bounded by the respectively m(b − a) and M(b − a) that is follows as
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Inequalities between functions.
If f(x) ≤ g(x) for each x in [a, b] then each of the upper and lower sums of f is bounded above by the upper and lower sums respectively of g. Thus
Thus there is a generalization of the above inequalities as M(b − a) is the integral of the constant function with value M over [a, b].
In this a [c, d] is a subinterval of [a, b] and f(x) is non-negative for all x, then
APPLICATIONS OF THE DEFINITE INTEGRAL
The definite integral of a function are used in many applications. Those discussed here
In this by dividing the interval up into n equal subintervals each of width ï„x and we will denote the point on the curve at each point by Pi than we can then approximate the curve by a series of straight lines connecting the points.
Here is a sketch of this situation for n ï€½ï€ 9 .we can the length of each sub intervals and applying the mean value theorem to find the length of the arc.
Like as arc length we also find the surface area by dividing in to sub intervals.
The approximation on each interval gives a distinct portion of the solid and to make this clear each portion is colored differently. Each of these portions are called frustums .
Probability density functions satisfy the following conditions.
1. f ( x) ³ 0 for all x.
2 ò¥. f ( x)dx =1
Probability density functions can be used to determine the probability that a continuous randomvariable lies between two values, say a and b. This probability is denoted by P(a £ X £ b) and it is given by
In physics work is done when a force acting upon an object causes a displacement. For example, riding a bicycle.If the force is not constant then we must use integration to find the work done.
where F(x) is the variable force.
In this for v velocity in terms of t the time we can find the displacement( s) of a moving object from time t = a to time t = b by integration, as follows:
Area bounded by a curve
The definite integral only gives us an area when the whole of the curve is above the x-axis in the region from x = a to x = b. If this is not the case, we have to break it up into individual sections
In mathematics the trapezoidal rule is also known as the trapezoid rule and trapezium rule. It is an approximate technique for calculating the definite integral
The trapezoidal rule works by approximating the region under the graph of the function f(x) as a trapezoid and calculating its area. It follows that
It is a method of finding an approximate value for an integral that is based on finding the sum of the areas of trapezial. Suppose we wish to find an approximate value for ∫ab f(x)dx. The interval a≤x≤b is divided up into n sub-intervals, each of length h=(b − a)/n, and the integral is approximated by ½h(y0+2y1+2y2+...+2yn−1+yn)
This is the sum of the areas of the individual trapezial one of which is shown in the diagram. The error in using the trapezium rule is approximately proportional to 1/n2, so that if the number of sub-intervals is doubled, the error is reduced by a factor of 4.
In numerical analysis, Simpson's rule is a method for numerical integration the numerical approximation of definite integrals. It is the following approximation:
The method is formed by the mathematician Thomas Simpson of Leicestershire in England.In numerical analysis Simpson's rule is a method for numerical integration the numerical approximation of definite integrals. A basic approximation formula for definite integrals which states that the integral of a real-valued function ƒ on an interval [a,b] is approximated by h[ƒ(a) + 4ƒ(g + h) + ƒ(b)]/3
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where h = (b - a)/2
This is the area under a parabola which coincides with the graph of ƒ at the abscissas a, a + h, and b. A method of approximating a definite integral over an interval which is equivalent to dividing the interval into equal subintervals and applying the formula in the first definition to each subinterval
Comparision trapezoidal rule and simpson rule.
The trapezoidal rule and Simpson's rule are for continuous
Functions for instance, functions which are the continuous of bounded variation or
which are absolutely continuous and whose derivative is in Lp. These differ considerably from the classical results which require the functions to have continuous higher derivatives.The results are sharp in both cases .In many cases precisely characterize the functions for
which equality its holds. One consequence of these results is that the functions the error estimates for the trapezoidal rule are better than the Simpson's rule because it has have smaller constants . it is given a finite interval I = [a, b] and a continuous function f:I→R and there are two elementary methods for approximating the integral
The trapezoidal rule and Simpson's rule. Partition the interval I into n intervals of equal length with endpoints xi = a +i|I|/n, 0≤ i≤ n. Then the trapezoidal rule approximates the integral with the sum.
Similarly, if we partition I into 2nintervals, Simpson's rule approximates the integral with the sum
Both approximation methods have well-known error bounds in terms of higher derivatives:
In this it is estimates that the derived using polynomial approximation which leads naturally to the higher derivatives on the righthand sides. However there is a assumption that f is not only continuous but it must have continuous higher order derivatives which means that we cannot use them to estimate directly the error when approximating the integral of such a well-behaved function as
f(x) =√x on [0, 1]. It is possible to use them indirectly by approximating f with a smooth function
A comparison of the pseudocodes for the trapezoidal and Simpson's rules shows that the Simpson's code is only slightly more complex than the trapezoidal code. Most importantly, each requires the same number of function for evaluations . The trapezoidal rule was derived by approximating the integrand by a piecewise-linear function. The integral is a linear function then the trapezoidal rule gives an exact result. Since Simpson's rule was derived by approximating the integrand by a piecewise-quadratic function, we expect an exact result if the integrand is a quadratic function. But Simpson's does an even better job by a fortunate accident it turns out that Simpson's rule gives an exact result for integrating a cubic function. So if the integrand is a well-behaved function then Simpson's rule is to be preferred. The only other possible drawback is the requirement that n be even this requirement is rarely of any consequence