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Double Integrals
Double Integrals
Before starting on double integrals let’s do a quick review of the definition of a definite integrals
for functions of single variables
...
For these integrals we can say that we are
integrating over the interval a ≤ x ≤ b
...
Now, when we derived the definition of the definite integral we first thought of this as an area
problem
...
A ≈ f ( x0* ) Δx + f ( x1* ) Δx +
+ f ( xi* ) Δx +
+ f ( xn* ) Δx
To get the exact area we then took the limit as n goes to infinity and this was also the definition of
the definite integral
...
With functions of one
variable we integrated over an interval (i
...
a one-dimensional space) and so it makes some sense
2
then that when integrating a function of two variables we will integrate over a region of
(twodimensional space)
...
Also, we will initially assume that f ( x, y ) ≥ 0 although this doesn’t really have to be the case
...
Now, just like with functions of one variable let’s not worry about integrals quite yet
...
We will first approximate the volume much as we approximated the area above
...
This will
divide up R into a series of smaller rectangles and from each of these we will choose a point
( x , y )
...
*
i
*
j
Now, over each of these smaller rectangles we will construct a box whose height is given by
f ( xi* , y *j )
...
(
)
Each of the rectangles has a base area of Δ A and a height of f xi* , y *j so the volume of each of
(
)
these boxes is f xi* , y *j Δ A
...
To get a better estimation of the volume we will take n and m larger and larger and to get the
exact volume we will need to take the limit as both n and m go to infinity
...
This looks a lot like the definition of the integral of a function of
single variable
...
Here is the official definition of a double integral of a function of two variables over a rectangular
region R as well as the notation that we’ll use for it
...
We have two integrals to
denote the fact that we are dealing with a two dimensional region and we have a differential here
as well
...
Note
as well that we don’t have limits on the integrals in this notation
...
Note that one interpretation of the double integral of f ( x, y ) over the rectangle R is the volume
under the function f ( x, y ) (and above the xy-plane)
...
We can do this by choosing xi* , y *j to be the midpoint of each rectangle
...
This leads to the Midpoint Rule,
∫∫ f ( x, y ) dA ≈ ∑∑ f ( x , y ) Δ A
n
R
m
i =1 j =1
i
j
In the next section we start looking at how to actually compute double integrals