%
% Function mfile lint.m
%
% This mfile estimates the line integral of a two-dimensional vector
% field F = [u(x,y), v(x,y)] along a path C. The call is
% lint(u,v,corners) where $u$ and $v$ are functions of x,y given
% in mfiles or as inline functions. corners = [a b c d]
% is the vector of corner coordinates of a rectangle R.
% After the call, user is asked to enter the number of line
% segments in the path. User then clicks with left mouse
% button on the figure, until the number of segments is completed
% The value of the line integral on each segment is displayed on
% the screen.
% The program also estimates the integral of xdy along C.
% When C is a closed polygon, this line integral is the area of
% the enclosed polygon.
function out = lint(u,v,corners)
a = corners(1); b = corners(2); c = corners(3); d = corners(4);
x = linspace(a,b,11);
y = linspace(c,d,11);
[X,Y] = meshgrid(x,y);
U = feval(u,X,Y);
V = feval(v,X,Y);
quiver(X,Y,U,V)
axis image
hold on
N = input('enter the number of segments in your path ')
linesum = 0;
A = 0;
[x0,y0] = ginput(1);
n = 50;
simpvec = 2*ones(1,n+1);
simpvec(2:2:n) = 4*ones(1,n/2);
simpvec(1) = 1; simpvec(n+1) = 1;
for j = 1:N
[x1, y1] = ginput(1);
plot([x0,x1], [y0,y1], 'r')
xm = .4*x0 + .6*x1; ym = .4*y0 + .6*y1;
d = norm([x1-x0, y1-y0]);
l = .025*max([b-a, d-c]);
delx = (l/d)*(x1-x0); dely = (l/d)*(y1-y0);
xedge = [xm, xm-delx-dely, xm-delx+dely];
yedge = [ym, ym-dely+delx, ym-dely-delx];
fill(xedge, yedge, 'r')
segment_number = num2str(j);
text(x1,y1, segment_number)
dx = (x1-x0)/n; dy = (y1-y0)/n;
xx = linspace(x0, x1,n+1);
yy = linspace(y0 ,y1,n+1);
Fdx = simpvec*feval(u,xx,yy)'*dx/3;
Fdy = simpvec*feval(v,xx,yy)'*dy/3;
line_integral = Fdx + Fdy
linesum = linesum + line_integral;
dA = .5*(x1+x0)*(y1-y0);
A = A +dA;
x0 = x1; y0 = y1;
end
area = A
total_line_integral = linesum
hold off