Math 241 - Quiz #11

12.6 #9 Find the area of the surface with parametric equations
  x = uv
  y = u + v
  z = u - v
  u^2 + v^2 <= 1
(in class I switched the equations for y and z)

Solution

r_u = < v, 1, 1>
r_v =
r_u x r_v = <-2, u+v, v-u>
|r_u x r_v| = sqrt(4 + (u+v)^2 + (v-u)^2)
  = sqrt(4 + 2u^2 + 2v^2)

Surface Area = int int sqrt(4+2u^2+2v^2) du dv

Because of the domain and the form of the integrand, we change to polar coordinates, with $u^2+v^2 = r^2$, and get

SA = int_0^2pi int_0^1 sqrt(4+2r^2) r dr dtheta
  = pi/3(6sqrt(6)-8)

ccollins@math.utk.edu
Last Modified: Apr. 5, 1999