# Let F(x, y, z) = (y, -3x, y z) and G = F. Calculate the surface integral I =  G dS, where the surface S is the triangular portion of the plane defined by the three vertices (x, y, z) = (2, 0,...

Let F(x, y, z) = (y, -3x, y z) and G = F. Calculate the surface integral I =  G dS, where the surface S is the triangular portion of the plane defined by the three vertices (x, y, z) = (2, 0, 0), (x, y, z) = (0, 1, 0), (x, y, z) = (0, 0, 4).
a) Sketch the surface S and write the cartesian equation of the plane.
b) Write a parametrization of the surface S.
c) Calculate I.

I suppose we start with the equation for I and use the equation for G?

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Borys Shumyatskiy | Certified Educator

Denote the given points:

First, find the unit vector orthogonal to (for a plane it is a constant vector). An orthogonal vector is

Its unit vector is

a)

Therefore the equation of the plane is

or

b) The parametrization of is

Therefore

c) To find the integral, find first:

Then compute the dot product

To finally find the surface integral, note that on the -plane under the variable  is from to

and the variable is from to

Also note that and the surface integral becomes

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