Calculate area from plot

I have this plot and I was wondering if there's an easy way to calculate the total area that is coloured red. I can sum the areas of the circles but then I get overlapping pieces which I have no idea how to calculate and substract from the total.

Image

Answer 1

[Grzegorz Knor]

Simple idea (not very accurate): test:

t = 0:.01:2*pi;
hold on
P = 0;
for k=1:10
    x0 = k^2;
    y0 = 10*rand;
    r = sqrt(k);
    x = r*sin(t)+x0;
    y = r*cos(t)+y0;
    fill(x,y,'r','EdgeColor','none')
    P = P + pi*r*r;
end
axis equal
a = getframe(gca);
a = a.cdata;
diff(get(gca,'ylim'))*diff(get(gca,'xlim'))*sum(sum(a(:,:,1)==255&a(:,:,2)==0&a(:,:,3)==0))/(size(a,1)*size(a,2))
P

overlapping circles:

t = 0:.01:2*pi;
hold on
for k=1:10
    x0 = 10*rand;
    y0 = 10*rand;
    r = sqrt(k);
    x = r*sin(t)+x0;
    y = r*cos(t)+y0;
    fill(x,y,'r','EdgeColor','none')
end
axis equal
a = getframe(gca);
a = a.cdata;
diff(get(gca,'ylim'))*diff(get(gca,'xlim'))*sum(sum(a(:,:,1)==255&a(:,:,2)==0&a(:,:,3)==0))/(size(a,1)*size(a,2))