Surface integrals using an explicit description. evaluate the surface integral \iint_{s}^{}f(x, y,z)ds using an explicit representation of the surface.

f(x, y,z) = ez; s is the plane z = 8 - x - 2y in the first octant.

Parameterize by the vector function

so that the normal vector to is given by

with magnitude

In this case, the normal vector is

with magnitude . The integral of over is then

where is the region in the plane over which is defined. In this case, it's the triangle in the plane which we can capture with and , so that we have

y = 3x+4

step-by-step explanation:

to find the slope

slope = (y2-y1)/ (x2-x1)

    = (-8-4)/ (-4-0)

    = -12/-4

  = 3

using the slope intercept form of a line

y= mx+b

we know the slope is 3

we know the y intercept is 4 ( the point where it crosses the y axis or x=0)

y = 3x+4