double integration of parametric function

hello all,

I know how to plot a parametric surface, for example as in

syms u v
x = u * cos(v);
y = u * sin(v);
z = v;
fsurf(x, y, z, [0 5 0 4*pi])

but can someone point me to the appropriate function for the double integration that calculates the surface area, for example over the interval

0 <= u <= 5
0 <= v <= 4*pi

of the example above?

regards, Danny.

Answer 1

[David Goodmanson]

Hi Danny,

You can find the surface area by finding the vectors Du and Dv that are parallel to the surface when you vary u and v respectively. Taking their cross product gives the the normal unit vector n, times the area element dS of a parallelogram whose area is proportional to dudv. Integrating the area elements give the total area. Since the area element does not depend on v, you can multiply by 4*pi and just do the u integral.

This procedure is analogous to finding the Jacobian. It's almost easier to do this by hand than to use syms, but

syms u v a real
x = u * cos(v);
y = u * sin(v);
z = a*v;         % in case you want to vary the pitch
zplot = v;
fsurf(x, y, zplot, [0 5 0 4*pi])
Du = diff([x,y,z],u)     % Du is diff() times du
Dv = diff([x,y,z],v)     % Dv is diff() times dv
dS = simplify(norm(cross(Du,Dv)))
Area = 4*pi*int(dS,0,5)
double(subs(Area,a,1))


% results
Du = [ cos(v), sin(v), 0]
Dv = [ -u*sin(v), u*cos(v), a]
dS = (a^2 + u^2)^(1/2)
Area =  4*pi*((a^2*log((a^2 + 25)^(1/2) + 5))/2 - (a^2*log(a^2))/4 + (5*(a^2 + 25)^(1/2))/2)
ans =  174.7199

[Danny Van Elsen]

hi David, thanks for your answer!

Indeed, I have the same solution as

(manual)
r = cos(v)i + u*sin(v)j + vk

dr/du = cos(v)i + sin(v)j
dr/dv = u*sin(v)i + u*cos(v)j + k 

||dr/du X dr/dv|| = sqrt(sin(v)^2 + cos(v)^2 + u^2)

which then leads to the double integral

(matlab)
fun = @(u,v) sqrt(1 + u.^2)
q = integral2(fun, 0, 5, 0, 4*pi)

q =  174.7199

I can recognize the same elements in your answer.

But somehow, and probably naively, I was expecting there to be a function taking the r-vector and the limits as input, and then "just" calculate the area. In one line as it were. So this is probably not the case?

[David Goodmanson]

Hi Danny, I don't think so, although I would be happy to be proved wrong. For a volume integral you can at least get the Jacobian basically in one line with symbolic variables.

Answer 2

[Rajat Tewari]

You can refer to integral2 function of MATLAB to calculate the double integraltion and area.