Question

Session 48 Problems PDF: How did they know that the height of the surface was z = 1-x-y? I knew that the region R was bounded by y = -x+1, but how do you go from this equation to the z equation?

Answer 1

the slant face of the tetrahedron is a 'plane' through the points (0,0,1) (0,1,0) and (1,0,0)

so it has the equation in the form
ax + by +cz = d

substitute each of the three points and we get:
c=d
b=d
a=d

thus the equation is
dx + dy + dz=d

or

x+y+z=1

=> z=1-x-y

Answer 2

Another way, if you know cross products. if we call the 3 points a, b, c
then form the vectors (for example) b-a, and c-a
now form the cross product and you will find the vector normal to the plane

once we have the normal we solve for the constant:

N dot P = c

thus:
(0,0,1) - (0,1,0) = (0, -1, 1)
(1,0,0) - (0,1,0)= (1, -1, 0)
cross product of (0,-1,1) X (1, -1, 0) = (1, 1, 1)
now we can do
<1,1,1> dot <x,y,z> = c
put in any point on the plane, e.g. (1, 0 , 0)
we get 1+0+0= c
and the equation of the line is
x +y + z = 1
and finally
z= 1-x-y

more work than baru's approach (where we have lots of zeros), but it might be easier if we are given "harder" points.