The Chemistry Maths Book, Second Edition

9.10 The double integral 283

9.10 The double integral

The double integral can be defined as the limit of a (double) sum in the same

way as the Riemann integral was defined in Section 5.4. Letf(x,y)by a continuous

function of xand yin a rectangular region of the xy-plane, fora 1 ≤ 1 x 1 ≤ 1 bandc 1 ≤ 1 y 1 ≤ 1 d

(Figure 9.12).

Divide the intervalx 1 = 1 a 1 → 1 binto msubintervals of width∆x

r

1 = 1 x

r

1 − 1 x

r− 1

and the

intervaly 1 = 1 c 1 → 1 dinto nsubintervals of width∆y

s

1 = 1 y

s

1 − 1 y

s− 1

; that is, divide the

rectangle into small rectangles of area∆A

rs

1 = 1 ∆x

r

∆y

s

. The integral (9.53) is then

defined as the limit

(9.54)

(when the limit exists).

The double summation (9.54) can be performed in either order: either as

(9.55)

wherev

r

is the sum over the elements in a vertical strip and the second sum is over

the vertical strips, or as

(9.56)

whereh

s

is the sum over the elements in a horizontal strip and the second sum is over

the horizontal strips. These two orders of summation correspond to the two orders of

integration in (9.53).

Just as the integral in one variable was interpreted in Section 5.4 as the area under

the curve, the double integral can be interpreted as the ‘volume under the surface’;

fx y x y h y

rs r

r

m

s

s

n

s

(),=

=





























=

∑∑

∆∆ ∆

11

ss

s

n

=

∑

1

fx y y x x

rs s

s

n

r

r

m

r

(),=

=



























= 

∑∑

∆∆ ∆

11

v

rr

r

m

=

∑

1

ZZ

c

d

a

b

rs r s

r

fxydxdy fx y x y

m

n

( ),=lim ( ,)

→

→

=

∞

∞

∆∆

1

mm

s

n

∑∑

= 1

y

x

a

b

∆ x

r

c

d

∆ y

s

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Figure 9.12