example of integration with respect to surface area on a helicoid

In this example, we shall consider itegration with respect to surface area on the helicoid.

The helicoid may be parameterized as follows:

$$x=u\sin v$$

$$y=u\cos v$$

$$z=cv$$

(The constant $c$ may be thought of as the “pitch of the screw”.) Computing derivatives and appying trigonometric identities, we obtain

From this we have

$$\sqrt{{\left(\frac{\partial (x,y)}{\partial (u,v)}\right)}^2+{\left(\frac{\partial (y,z)}{\partial (u,v)}\right)}^2+{\left(\frac{\partial (z,x)}{\partial (u,v)}\right)}^2}=$$

$$\sqrt{u^2+c^2{\cos }^{2}v+c^2{\sin }^{2}v}=\sqrt{u^2+c^2}$$

so we can compute area integrals over helicoids as follows

$${\int }_{S}f(u,v)d^{2}A=\int f(u,v)\sqrt{c^2+u^2}𝑑u𝑑v$$

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