Matlab: Area between closed curve and a line

Question

what is the best way to calculate the upper area between the closed curve and the line?

alpha = [0:pi/360:2*pi];
x = 2.*(1-cos(alpha)).*cos(alpha);
y = 2.*(1-cos(alpha)).*sin(alpha);
w = [-4:0.1:0.5];
z = 0.5*w;
plot(x,y);
hold on;
grid on;
plot(w,z);

Is it possible to solve it by using trapz, even if it is a closed curve and occupying both positive and negative y axis? I saw a lot of topics about just a closed curve or a curve and a line. But none about a clsed curve and a line.

Thanks.

Answer 1

You just need to use interp1 to calculate the area just for the range you want. Firstly you calculate the area between the curve and the x axis (over):

a1 = abs(trapz(x,y)/2);

Then the area between the line and the x axis (under) for x going from the intersection point to zero:

x3 = [-3.38 0];
r1 = interp1(x2,y2,x3);
a2 = abs(trapz(x3,r1));

And then the same thing for the rest of the curve under the x axis:

x4 = [-3.99 -3.38];
r2 = interp1(x,y,x4);
a3 = abs(trapz(x4,r2))/2;

And the total area would be:

at = a1 + a2 + a3;

And here the whole Code:

clear all;
clc;

z = 0:0.01:2*pi;

x = 2.*(1-cos(z)).*cos(z);
y = 2.*(1-cos(z)).*sin(z);

plot(x,y);
hold on;
grid on;

x2 = -4:0.01:0.5;
y2 = 0.5*x2;

plot(x2,y2);

a1 = abs(trapz(x,y)/2);

x3 = [-3.38 0];
r1 = interp1(x2,y2,x3);
a2 = abs(trapz(x3,r1));

x4 = [-3.99 -3.38];
r2 = interp1(x,y,x4);
a3 = abs(trapz(x4,r2))/2;

at = a1 + a2 + a3;