Multivariate Calculus - Bruce E. Shapiro

156 LECTURE 19. SURFACE AREA WITH DOUBLE INTEGRALS
the yz plane with the tangent plane; the slope of this line is therefore
m = f y (x, y)
The vector v from (x, y, z) to (x, y + ∆y, z + m∆y) is
Figure 19.2: Calculation of vectors that edge the infinitesimal tangent plane.
v = (0, ∆y, m∆y) = (0, ∆y, f y (x, y)∆y)
Similarly, the left edge of the parallelogram is formed by the intersection of a plane
parallel to the xz-plane with the tangent plane; the slope of this line is therefore
m x = f x (x, y)
By a similar construction, the vector u from (x, y, z) to (x + ∆x, y, z + m x ∆x) is
u = (∆x, 0, m x ∆x) = (∆x, 0, f x (x, y)∆x)
The area of the parallelogram formed by u and v is given by their cross product:
∆A = ‖u × v‖
⎛
⎞ ⎛ ⎞
0 −f x (x, y)∆x 0 0
=
⎝
f x (x, y) 0 −∆x⎠
⎝ ∆y ⎠
∥ 0 ∆x 0 f y (x, y)∆y ∥
⎛
⎞
−f x (x, y)∆x∆y
=
⎝
−f y (x, y)∆x∆y⎠
∥ ∆x∆y ∥
√
= ∆x∆y 1 + fx(x, 2 y) + fy 2 (x, y)
Taking the limit as the size of the infinitesmal parallelograms go to zero, we have
√
dA = dA = 1 + fx(x, 2 y) + fy 2 (x, y) dxdy
Theorem 19.1 . Suppose that z = f(x, y) : D ∪ R 2 ↦→ R Then the surface area of
f over D is
√
A = 1 + fx(x, 2 y) + fy 2 (x, y) dxdy
D
Revised December 6, 2006. Math 250, Fall 2006