where
E=∂r
∂u
·∂r
∂u
=
(
∂x
∂u
) 2
+
(
∂y
∂u
) 2
+
(
∂z
∂u
) 2
F=∂r
∂u
·∂r
∂v
=∂x
∂u
∂x
∂v
+∂y
∂u
∂y
∂v
+∂z
∂u
∂z
∂v
G=∂r
∂v
·∂r
∂v
=
(
∂x
∂v
) 2
+
(
∂y
∂v
) 2
+
(
∂z
∂v
) 2
.
Then the surface area can be represented in the form
S=
∫
Ruv
√
EG −F^2 du dv, (7 .104)
where the integration is over those parameter values uand vwhich define the surface.
The various surface integrals can also be represented in terms of the parameters
uand v. These integrals have the forms
∫∫
R
f(x, y, z)dS=
∫∫
Ruv
f(x(u, v ), y (u, v ), z(u, v ))
√
EG −F^2 ˆendu dv
and
∫∫
R
F(x, y, z)·dS=
∫∫
Ruv
F(x(u, v ), y (u, v ), z (u, v )) ·eˆn
√
EG −F^2 du dv.
(7 .105)
Example 7-33. A cylinder of radius a and height h has the parametric
representation x =x(θ, z ) = acos θ, y =y(θ, z ) = asin θ, z =z(θ, z ) = z, where the
parameters θand z, are illustrated in figure 7-24, and satisfy 0 ≤θ≤ 2 πand 0 ≤z≤h.
Figure 7-24.
Surface area in cylindrical coordinates.
A point on the surface of the cylinder can be represented by the position vector
r =r (θ, z) = acos θˆe 1 +asin θˆe 2 +zˆe 3.
