Advanced Calculus 201-BNK-05
Vincent Carrier
Surface Area (I)
Let f : R2 → R and consider a region E ⊂ R2 of the xy-plane. Assume that f (x, y) ≥ 0
for all (x, y) ∈ E. Let A be the area of the surface defined by the graph of f over E.
z 6
z 6
A
PP
PP
)
PP
P
PP
PPP
PP
P
x
PP
P
P
PP
PP
PP
PP
)
q
PP
P
PP
q
P
y
PP
P
x
y
To make matters simpler, assume that E is of the form [a, b] × [c, d]. Let us consider
partitions {x0 , x1 , x2 , . . . , xm } of [a, b] and {y0 , y1 , y2 , . . . , yn } of [c, d] such that
a = x0 < x 1 < x 2 < · · ·
< xm−1 < xm = b
c = y0 < y1 < y2 < · · ·
< yn−1
< yn
= d
with
∆x =
b−a
m
and
∆y =
d−c
.
n
(x∗i , yj∗ ) = (xi , yj )
d
r
∆y
∆A = ∆x ∆y
c
a
∆x
b
Let us take (x∗i , yj∗ ) = (xi , yj ) for i = 1, 2, . . . , m and j = 1, 2, . . . , n. Consider a subrectangle of the partition and the portion of the tangent plane to the surface of the graph
of f at (xi , yi ) over the sub-rectangle.
(xi , yi , f (xi , yi ))
~u
r
XXXXX
z
+
XXX
XX
~v
(x , y )
r i i
PPP
P
P
PP
PPP
∆y
∆x
~u = (∆x, 0, fx (xi , yi ) ∆x)
~v = (0, ∆y, fy (xi , yi ) ∆y)
The area of the tangent plane above the sub-rectangle is
k~u × ~v k = k(−fx (xi , yi ) ∆x ∆y, −fy (xi , yi ) ∆x ∆y, ∆x ∆y)k
q
=
[fx (xi , yi ) ∆x ∆y]2 + [fy (xi , yi ) ∆x ∆y]2 + (∆x ∆y)2
=
q
[fx (xi , yi )]2 + [fy (xi , yi )]2 + 1 ∆x ∆y
=
q
[fx (x∗i , yi∗ )]2 + [fy (x∗i , yi∗ )]2 + 1 ∆A
and thus
A =
lim lim
m→∞ n→∞
m X
n q
X
[fx (x∗i , yi∗ )]2 + [fy (x∗i , yi∗ )]2 + 1 ∆A
i=1 j=1
ZZ q
=
[fx (x, y)]2 + [fy (x, y)]2 + 1 dA.
E
Therefore,
ZZ q
A =
[fx (x, y)]2 + [fy (x, y)]2 + 1 dA.
E