HOW A LINE INTEGRAL GENRALISES A DEFINIT INTEGRAL KNOWN FROM CALCULUS

: HOW A LINE INTEGRAL GENRALISES A DEFINIT INTEGRAL KNOWN FROM CALCULUS





Submitted to: Submitted by:
Mr. Brijesh kumar Sinha Yuvraj Singh
Deptt. Of Mathematics RB1803B11
10809366
B.Tech-MBA(cse)


ACKNOWLEDGEMENT


I, express my gratitude towards our subject teacher for the guidelines and help provided by her in making this project a success. She helped me a lot in completing this project.
I would like to say thank you to all those who are involved in this project including my friends. Their valuable inputs in various matter related to the topic helped me a lot.
I have taken the help of many books and websites, listed in references. I would like to thank the library of the university that acted as a database of knowledge for me.
The various sites visited by me on the internet also helped me a lot in making my term paper a success. I thank again one and all.

CONTENTS

1. Line Integral
2. Definite Integral
3. Area under a curve
4. Area using line integral
5. Application of line integral
6. References.


LINE INTEGRAL



The line integral of a vector field on a curve is defined by

(1)
where denotes a dot product. In Cartesian coordinates, the line integral can be written

(2)
where

(3)
For complex and a path in the complex plane parameterized by ,

(4)
Poincaré's theorem states that if in a simply connected neighborhood of a point , then in this neighborhood, is the gradient of a scalar field ,

(5)
for , where is the gradient operator. Consequently, the gradient theorem gives

(6)
for any path located completely within , starting at and ending at .
This means that if (i.e., is an irrotational field in some region), then the line integral is path-independent in this region. If desired, a Cartesian path can therefore be chosen between starting and ending point to give

(7)
If (i.e., is a divergenceless field, a.k.a. solenoidal field), then there exists a vector field such that

(8)
where is uniquely determined up to a gradient field (and which can be chosen so that ).
.
DEFINITE INTEGRAL



A definite integral is an integral

(1)
with upper and lower limits. If is restricted to lie on the real line, the definite integral is known as a Riemann integral (which is the usual definition encountered in elementary textbooks). However, a general definite integral is taken in the complex plane, resulting in the contour integral

(2)
with , , and in general being complex numbers and the path of integration from to known as a contour.
The first fundamental theorem of calculus allows definite integrals to be computed in terms of indefinite integrals, since if is the indefinite integral for a continuous function , then

(3)
This result, while taught early in elementary calculus courses, is actually a very deep result connecting the purely algebraic indefinite integral and the purely analytic (or geometric) definite integral. Definite integrals may be evaluated in Mathematica using Integrate[f, x, a, b ].

AREA UNDER A CURVE
Theory:
The definite integral can be used to find the area between a graph curve and the ‘x’ axis, between two given ‘x’ values. This area is called the ‘area under the curve’ regardless of whether it is above or below the ‘x’ axis.
When the curve is above the ‘x’ axis, the area is the same as the definite integral ...

but when the graph line is below the ‘x’ axis, the definite integral is negative. The area is then given by:

Sometimes part of the graph is above the ‘x’ axis and part is below, then it is necessary to calculate several integrals. When the area of each part is found, the total area can be found by adding the parts.

For example, to find the area between the graph of: y = x² - x - 2 and the ‘x’ axis, from x = -2 to x = 3, we need to calculate three separate integrals:

The zeros of the function f(x) that lie between -2 and 3 form the boundaries of the separate area segments.
In this case there are zeros at x = -1 and x = 2, (see graph above) and so three separate areas must be found: A1, A2 and A3 as follows:

So the total shaded area between the function and the graph from x = -2 to x = 3 is given by:
A = A1 + A2 + A3
Now we can graph the function, locate the zeros and calculate the definite integrals.
AREA USING LINE INTEGRAL
The stated proposition is: if a given region is bounded by a piecewise smooth closed orientable curved then the area is given as a line integral over the curve. We work through examples where both a parameterization is given and where a parameterization will be determined. These examples show that finding area with line integrals can be straightforward.
Proposition (Line Integral for Area) If is a region bounded by a piecewise smooth simple closed curve oriented counterclockwise, then the area of is given by

















EXAMPLE

(Line Integral for Area) Use a line integral to find the area enclosed by the region defined by the circle

Solution. We can parametrize the circle by and for Then the area is found by,






Use a line integral to find the area enclosed by the region defined by the triangle with vertices

Solution. We can parametrize the line segments by







Then the area is found by,












APPLICATION OF LINE INTEGRAL
Applications
The line integral has many uses in physics. For example, the work done on a particle traveling on a curve C inside a force field represented as a vector field F is the line integral of F on C.
Complex line integral
The line integral is a fundamental tool in complex analysis. Suppose U is an open subset of C, γ : [a, b] → U is a rectifiable curve and f : U → C is a function. Then the line integral

may be defined by subdividing the interval [a, b] into a = t0 < t1 < ... < tn = b and considering the expression

The integral is then the limit of this sum, as the lengths of the subdivision intervals approach zero.
If γ is a continuously differentiable curve, the line integral can be evaluated as an integral of a function of a real variable:

When γ is a closed curve, that is, its initial and final points coincide, the notation

is often used for the line integral of f along γ.
The line integrals of complex functions can be evaluated using a number of techniques: the integral may be split in to real and imaginary parts reducing the problem to that of evaluating two real-valued line integrals, the Cauchy integral formula may be used in other circumstances. If the line integral is a closed curve in a region where the function is analytic and containing no singularities, then the value of the integral is simply zero, this is a consequence of the Cauchy integral theorem. Because of the residue theorem, one can often use contour integrals in the complex plane to find integrals of real-valued functions of a real variable






REFERENCE

www.wikipedia.com
www.mathworld.com
www.about.com
www.britannica.com
Engineering Mathematics- H.K.Das
Encarta Encyclopedia