double integration of parametric function

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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.

采纳的回答

David Goodmanson
David Goodmanson 2020-4-15
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
  2 个评论
Danny Van Elsen
Danny Van Elsen 2020-4-16
编辑:Danny Van Elsen 2020-4-16
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
David Goodmanson 2020-4-17
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.

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更多回答(1 个)

Rajat Tewari
Rajat Tewari 2020-4-13
You can refer to integral2 function of MATLAB to calculate the double integraltion and area.

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