Evaluate the surface integral. s y ds, s is the helicoid with vector equation r(u, v) = u cos(v), u sin(v), v , 0 ≤ u ≤ 6, 0 ≤ v ≤ π.

The surface integral (y ds), s is the helicoid with vector equation r(u,v) = u cos(v), u sin(v), v , 0 ≤ u ≤ 6, 0 ≤ v ≤ π is and this can be determine by doing the integration.

Given :

The surface integral (y ds), s is the helicoid with vector equation r(u, v) = u cos(v), u sin(v), v , 0 ≤ u ≤ 6, 0 ≤ v ≤ π.

The following calculation can be used in order to evaluate the surface integral.

The surface integral is given by:

 --- (1)

where and is given by:

Now, evaluate .

Now, substitute the known values in the expression (1).

Now, integrate the above expression.

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Compute the surface element:

So the integral is

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