Evaluate the surface integral. s xy ds s is the triangular region with vertices (1, 0, 0), (0, 8, 0), (0, 0, 8)

In this exercise we have to use the knowledge of parameterization and integrals to calculate the area needed, so we will find that this corresponds to:

Parameterizing the function given as:

Knowing that the integration limits will be given by:

So in this way calculating the surface we will find that it will be:

Now putting all this data together in the integral, we will find that it will correspond to:

See more about parameterizing at link

Parameterize by

with and .

Then the surface element is

and the surface integral is

a reflection across the x-axis , followed by the by a reflection across the y-axis , and then a scale dilation of 0.5 .

δabc ≅ δdcb by sas because

db ≅ ac . . given (side)

bc ≅ bc . . reflexive property (side)

∠dbc ≅ ∠ acb . . alternate interior angles where transversal bc crosses parallel lines bd and ac (included angle)