bài giảng vật lý bằng tiếng anh iteratedintegrals

MULTIPLE INTEGRALS

2.2
Iterated 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:

∫

d

c

f ( x, y ) dy

 x is held fixed
 f(x, y) is integrated with respect to y
from y = c to y = d


PARTIAL INTEGRATION

This procedure is called partial integration
with respect to y.

Notice its similarity to partial
differentiation.


PARTIAL INTEGRATION
d

Now, ∫c f ( x, y ) dy is a number that depends on
the value of x.

So, it defines a function of x:
d

A( x) = ∫ f ( x, y ) dy
c

Equation 1

PARTIAL INTEGRATION

If we now integrate the function A
with respect to x from x = a to x = b,
we get:

∫

b

a

A( x) dx = ∫  ∫ f ( x, y ) dy  dx

a 
 c
b

d


ITERATED INTEGRAL

The integral on the right side of Equation 1 is
called an iterated integral.

Equation 2

ITERATED INTEGRALS

Thus,
b

d

a

c

∫∫

f ( x, y ) dy dx = ∫  ∫ f ( x, y ) dy  dx

a 
 c
b

d

means that:
 First, we integrate with respect to y from c to d.
 Then, we integrate with respect to x from a to b.


ITERATED INTEGRALS

Similarly, the iterated integral
d

b

c

a

∫ ∫

f ( x, y ) dy dx = ∫

d

c

 b f ( x, y ) dx  dy
 ∫a


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.


ITERATED INTEGRALS

Example 1


Theorem 4

FUBUNI’S THEOREM

If f is continuous on the rectangle
R = {(x, y) |a ≤ x ≤ b, c ≤ y ≤ d}
then
b

d

a

c

∫∫ f ( x, y) dA = ∫ ∫
R

=∫

d

c

∫

b

a

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

a

∫∫ f ( x, y) dA = ∫ ∫
R

g ( x)h( y ) dx dy

= ∫  ∫ g ( x)h( y ) dx  dy

c 
 a
d

b


ITERATED INTEGRALS

In the inner integral, y is a constant.
So, h(y) is a constant and we can write:

∫

d

c

 b g ( x)h( y ) dx  dy = d  h( y )
∫c 
 ∫a


(∫

b

a

b

d

a

c

)


g ( x) dx dy


= ∫ g ( x) dx ∫ h( y ) dy

since

∫

b

a

g ( x) dx
is a constant.


Equation 5

ITERATED INTEGRALS

Hence, in this case, the double integral of f can
be written as the product of two single integrals:
b

d

a

c

g
(
x
)
h
(
y

)
dA
=
g
(
x
)
dx
h
(
y
)
dy
∫∫
∫
∫
R

where R = [a, b] x [c, d]


ITERATED INTEGRALS

Example 4