Plotting a parametric surface where the domain is non-rectangular

Hey I'm teaching a calculus course, and for an example in my lecture on surface integrals I would like to generate a surface plot in MATLAB of a portion of a circular cylinder. Here is my MATLAB code for the case where the parameter domain is rectangular:

clc; clear  
syms theta z  
x(theta,z) = cos(theta)
y(theta,z) = sin(theta)  
z(theta,z) = z  
fsurf(x, y, z, [0 2*pi 0 1])

However, the surfaced required for the problem instead has 0<z<1+sin(theta)

I'm not sure what is the best way to modify this code for the case where the domain is not rectangular. Thanks!

You could continue to use a rectangular domain, but use a dummy variable t and a mapping to your z:

syms theta t
x(theta,t) = cos(theta);
y(theta,t) = sin(theta);
z(theta,t) = t .* (1 + sin(theta));
fsurf(x, y, z, [0 2*pi 0 1])

You can do it numerically with the same result:

theta = linspace(0, 2*pi, 100);
t = linspace(0, 1, 50)';
X = repmat(cos(theta), length(t), 1);
Y = repmat(sin(theta), length(t), 1);
Z = t * (1 + sin(theta));
surf(X, Y, Z)

To avoid introducing t, you can do it by component with the same result:

theta = linspace(0, 2*pi, 100);
Nz = 50;
Z = nan(numel(theta),Nz);
for ntheta = 1:numel(theta)
    Z(ntheta,:) = linspace(0,1+sin(theta(ntheta)),Nz);
end
THETA = repmat(theta',1,Nz);
X = cos(THETA);
Y = sin(THETA);
surf(X, Y, Z)

The previous option is just a vectorized version of this.