Calculus 2nd Edition Rogawski

SECTION 16.2 Double Integrals over More General Regions 885
Equivalently, f is the value satisfying the relation
∫∫
f (x, y) dA = f · Area(D)
D
GRAPHICAL INSIGHT The solid region under the graph has the same (signed) volume as
the cylinder with base D of height f (Figure 13).
FIGURE 13
EXAMPLE 7 An architect needs to know the average height H of the ceiling of a
pagoda whose base D is the square [−4, 4] × [−4, 4] and roof is the graph of
H (x, y) = 32 − x 2 − y 2
where distances are in feet (Figure 14). Calculate H .
Solution First, we compute the integral of H (x, y) over D:
∫∫
∫ 4 ∫ 4
(32 − x 2 − y 2 )dA= (32 − x 2 − y 2 )dydx
D
−4 −4
∫ (
4
(
= 32y − x 2 y − 1 ) ∣ )
∣∣∣
4
3 y3 dx =
=
−4
( 640
3 x − 8 3 x3 ) ∣ ∣∣∣
4
−4
= 4096
3
The area of D is 8 × 8 = 64, so the average height of the pagoda’s ceiling is
∫∫
1
H =
H (x, y) dA = 1 ( ) 4096
= 64 ≈ 21.3 ft
Area(D) D
64 3 3
−4
∫ 4
−4
( 640
3 − 8x2 )
dx
The Mean Value Theorem states that a continuous function on a domain D must take
on its average value at some point P in D, provided that D is closed, bounded, and also
connected (see Exercise 63 for a proof). By definition, D is connected if any two points
in D can be joined by a curve in D (Figure 15).
P
P
Q
Q
(A) Connected domain: Any two
points can be joined by a curve
lying entirely in .
(B) Nonconnected domain.
FIGURE 15
FIGURE 14 Pagoda with ceiling
H (x, y) = 32 − x 2 − y 2 .
THEOREM 4 Mean Value Theorem for Double Integrals If f (x, y) is continuous and
D is closed, bounded, and connected, then there exists a point P ∈ D such that
∫∫
f (x, y) dA = f (P )Area(D) 9
D
Equivalently, f (P ) = f , where f is the average value of f on D.