multiple integrals 2.2 iterated integrals in this section, we will learn how to: express double...
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MULTIPLE INTEGRALS
2.2Iterated Integrals
In this section, we will learn how to:
Express double integrals as iterated integrals.
INTRODUCTION
Once we have expressed a double integral
as an iterated integral, we can then evaluate
it by calculating two single integrals.
INTRODUCTION
Suppose that f is a function of two variables that
is integrable on the rectangle
R = [a, b] x [c, d]
INTRODUCTION
We use the notation
to mean:
x is held fixed
f(x, y) is integrated with respect to y from y = c to y = d
( , )d
cf x y dy
PARTIAL INTEGRATION
This procedure is called partial integration
with respect to y.
Notice its similarity to partial differentiation.
PARTIAL INTEGRATION
Now, is a number that depends on the
value of x.
So, it defines a function of x:
( , )d
cf x y dy
( ) ( , )d
cA x f x y dy
PARTIAL INTEGRATION
If we now integrate the function A
with respect to x from x = a to x = b,
we get:
( ) ( , )b b d
a a cA x dx f x y dy dx
Equation 1
ITERATED INTEGRAL
The integral on the right side of Equation 1 is
called an iterated integral.
ITERATED INTEGRALS
Thus,
means that:
First, we integrate with respect to y from c to d. Then, we integrate with respect to x from a to b.
( , ) ( , )b d b d
a c a cf x y dy dx f x y dy dx
Equation 2
ITERATED INTEGRALS
Similarly, the iterated integral
means that:
First, we integrate with respect to x (holding y fixed) from x = a to x = b.
Then, we integrate the resulting function of y with respect to y from y = c to y = d.
( , ) ( , )d b d b
c a c af x y dy dx f x y dx dy
ITERATED INTEGRALS Example 1
FUBUNI’S THEOREM
If f is continuous on the rectangle
R = {(x, y) |a ≤ x ≤ b, c ≤ y ≤ d}
then
Theorem 4
( , ) ( , )
( , )
b d
a cR
d b
c a
f x y dA f x y dy dx
f x y dx dy
ITERATED INTEGRALS Example 2
ITERATED INTEGRALS Example 3
ITERATED INTEGRALS
To be specific, suppose that:
f(x, y) = g(x)h(y)
R = [a, b] x [c, d]
ITERATED INTEGRALS
Then, Fubini’s Theorem gives:
( , ) ( ) ( )
( ) ( )
d b
c aR
d b
c a
f x y dA g x h y dx dy
g x h y dx dy
ITERATED INTEGRALS
In the inner integral, y is a constant.
So, h(y) is a constant and we can write:
since is a constant.
( ) ( ) ( ) ( )
( ) ( )
d b d b
c a c a
b d
a c
g x h y dx dy h y g x dx dy
g x dx h y dy
( )
b
ag x dx
ITERATED INTEGRALS
Hence, in this case, the double integral of f can be
written as the product of two single integrals:
where R = [a, b] x [c, d]
Equation 5
( ) ( ) ( ) ( )b d
a cR
g x h y dA g x dx h y dy
ITERATED INTEGRALS Example 4