Line Integrals

 Line integral

§  Because the line integral needs to be computed repeatedly, it is advisable to encapsulate it within a function.

To create a function

·         The function should begin as follows:

function [y1, ..., yN] = fun_name(x1, ..., xM)

·         This declares a function named fun_name that accepts inputs x1, ..., xM and returns outputs y1, ..., yN. The function should end with end.

·         Function definition should be at the end of the main program.

·         To use the function, call it from the main main program as many times as needed.

%%%%%%%%%%%%%%%% Main Program %%%%%%%%%%%%%%%%%%

clc

clear

syms x y

LInt=0;

%Get the number of sub path in the curve C

n=input("How many subcurves are ther in the curve C? :")

%Calculate the line integral along each path

for i=1:n

    line(i)=LineIntegral(x,y);

    LInt =vpa(LInt +line(i))

end

%%%%%%%%%%%%%%%% Main Program ends %%%%%%%%%%%%%%%%%%

%Define function for Line integral

function line = LineIntegral(x,y)

option=input("1: y=f(x)\n2: x=g(y)\n")

 

switch option

    case 1

        y=input("Enter the function f(x): \n")

        r=[x y];

        F=input("Enter the vector function F(x,y)=[F1 F2]: F(x,y)")

        dr=diff(r,x);

        integrand1=dot(F,dr);

        xL=input("Enter the lower limit of x:");

        xU=input("Enter the Upper limit of x:");

        line=int(integrand1,x,xL,xU)

    case 2

        x=input("Enter the function g(y): \n")

        r=[x y];

        F=input("Enter the vector function F(x,y)=[F1 F2]: F(x,y)")

        dr=diff(r,y);

        integrand1=dot(F,dr);

        yL=input("Enter the lower limit of y:");

        yU=input("Enter the Upper limit of y:");

        line=int(integrand1,y,yL,yU)

    otherwise

        fprintf("Give the proper input")

end

 

end

%%%%%%%%%%%%%%%% Line Integral definition ends %%%%%%%%%%%%%%%%%%

Example 1

$$\text{Evaluate }\int F\cdot dr$$ $$\text{where c is the curve in plane }y=2x^2\text{ from }(0,0)\text{  to }(1,2)\text{  and }F=3xy\hat{i}-y^2 \hat{j}$$

How many subcurves are there in the curve C? :1

n =      1

1: y=f(x)

2: x=g(y)

1

option =      1

Enter the function f(x):

2*x^2

y =

2*x^2

Enter the vector function F(x,y)=[F1 F2]: F(x,y)=[3*x*y -y^2]

F =

[6*x^3, -4*x^4]

Enter the lower limit of x:0

Enter the Upper limit of x:1

line =

-7/6

LInt =

-1.1666666666666666666666666666667

Line integral using the parametric form

clear

clc

syms x y z t

x=input("Enter x(t):"); % If x ,y, z is a function of ‘t’

y=input("Enter y(t):");

z=input("Enter z(t):");

 

r=[x y z];

F=input("Enter the vector function F(x,y,z):")

dr=diff(r,t);

integrand1=dot(F,dr);

tl=input("Enter the lower limit of t:");

tu=input("Enter the Upper limit of t:");

line=int(integrand1,t,tl,tu)

Example 1

$$\text{Evaluate }\int F\cdot dr$$ $$\text{where c is the curve given by }x=2t^2, y=t, z=t^3$$

$$\text{from (0,0,0) to (2,1,1) and } F=(2y+3)\hat{i}+xz\hat{j}+(yz-x)\hat{k}.$$

Output

Enter x(t):2*t^2

Enter y(t):t

Enter z(t):t^3

Enter the vector function F(x,y,z):[2*y+3 x*z yz-x]

Enter the vector function F(x,y,z):[2*y+3 x*z y*z-x]

F =

[3 + 2*t, 2*t^5, - 2*t^2 + t^4]

Enter the Lower limit of t:0

Enter the Upper limit of t:1

line =

288/35