I was wondering how you would compute the surface area of a parameterized surface. Is there a formula or set of procedures you can follow to compute this. Say I wanted to compute the surface area of a portion of a helicoid. How would I accomplish this? I know that the parameterized form of a helicoid is x(u,v)=(ucos(v),usin(v),bv)
Would I just use this formula: Area = ∫ ∫ ||Tu x Tv(u,v)||dudv, Say I wanted the surface area over the region: 1 < u < 3, 0 <=v<=2
I am not sure what Tu and Tv(u,v) would be. Am I right in saying that Tu = cos(v), not sure about Tv(u,v) or what to do with bv in x(u,v). Thank you!
Define $$X(u,v) = (u\cos v, u \sin v, bv).$$
We compute its partial derivatives,
$$X_u(u,v) = (\cos v, \sin v, 0)$$
$$X_v(u,v) = (-u\sin v, u \cos v, 1)$$
The norm of the cross product is given by, $$\|X_u \times X_v \| = \sqrt{u^2 + 1}.$$
Therefore, your integral is now $$\int_{1}^{3} \int_{0}^{2} \sqrt{u^2 + 1} du dv = -\sqrt{2} + 3\sqrt{10} - \text{arcsinh}^{-1}(1) + \text{arcsinh}^{-1}(3). $$