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