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.

For more information, refer to the link given below:

link

Compute the surface element:

So the integral is

answer: the sun

explanation:

water cycle describes the evaporation of water from earth's surface, formation of clouds and precipitation water back onto the surface of earth. seventy percent of the earth is covered with water in the form of seas and oceans. the heat of the sun causes the evaporation of water from the surface of the ocean. the water vapors rise and condense to form water droplets. the collection of these water droplets form clouds which rain down again.

the evaporation of the oceans surface water as part of the water cycle is caused by the sun