Evaluate the Surface Integral over the Helicoid r(u,v) = ucos v i + usin v j + v k

Evaluate the surface integral ∫∫sqrt(1+x^2+y^2) dS over the helicoid r(u,v) = ucos v i + usin v j + v k.

Although this problem looks intimidating at first it's actually very manageable once we get into it.

The first thing to do for any surface integral is find the parameterization. In this problem, the parameterization is already given! (I know this is like stealing candy from a baby ... not that I eat much candy or steal from babies...)

Then if it's a scalar surface integral we take the magnitude of the normal vector |s_u x s_v|. This cross product acts like our jacobian for the change of variables.

In this problem our bounds are already given, but often we must find the bounds for the parameters next.

Lastly we substitute in our parameterization to a function, multiply by our Jacobian, then integrate.

As always, if you have any questions, let me know!

Thanks for watching!

-dr. dub

[omarato]

thank you but how is it a scalar function when we have parameterization in i j k ?