Multiple Integration
Recall our definition of the definite integral of a function of a single
variable:
Let f(x) be defined on [a,b] and let x0,x1,¼,xn
be a partition of [a,b]. For each [xi-1,xi], let xi*
Î [xi-1,xi]. Then
| |
ó
õ |
b
a
|
f(x) dx = |
lim
maxDxi
® 0
|
|
n
å
i = 1
|
f(xi*)Dxi |
|
We can extend this definition to define the integral of a function of two or
more variables.
Double Integral of a Function of Two Variables
Let f(x,y) be defined on a closed and bounded region R of the xy-plane. Set
up a grid of vertical and horizontal lines in the xy-plane to form an inner
partition of R into n rectangular subregions Rk of area
DAk, each of which lies entirely in R.
(Ignore the rectangles that are not entirely contained in R) Choose a point (xk*,
yk*) in each subregion Rk. The sum
| |
n
å
k = 1
|
f(xk*, yk*)
DAk |
|
is called a Riemann Sum. In the limit as we make our grid more
and more dense, we define the double integral of f(x,y) over R as
óó
õõ
R
|
f(x,y )dA = |
lim
maxDAk
® 0
|
|
n
å
k = 1
|
f(xk*, yk*)
DAk |
|
Notes
- If this limit exists, we say that f is integrable over the
region of integration R.
- If f is continuous on R, then f is integrable over R.
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