13.2 Iterated Integrals
Suppose that f (x, y) is an integrable over the rectangle R = [a, b] × [c, d].
ˆ
Partial integration
b
of f with respect to x:
f (x, y) dx
a
means that y is held xed and f (x, y) is integrated with respect to x from x = a to
x = b.
ˆ
Partial integration
d
of f with respect to y :
f (x, y) dy
c
means that x is held xed and f (x, y) is integrated with respect to y from y = c to
y = d.
Example 1.
ˆ
2
(a)
(x + 3y 2 ) dx
0
ˆ
(b)
3
exy dy
1
Iterated integrals
:
ˆ
dˆ b
ˆ
d ˆ b
f (x, y) dxdy :=
c
a
f (x, y) dx dy
c
a
means that we rst integrate with respect to x from a to b and then with respect to y
from c to d.
ˆ bˆ
ˆ b ˆ
d
f (x, y) dydx :=
a
c
d
f (x, y) dy dx
a
c
means that we rst integrate with respect to y from c to d and then with respect to x
from a to b.
Example 2. Evaluate the iterated integrals:
(a)
ˆ 4ˆ
1
(b)
√
x y dxdy
4
√
x y dydx
0
ˆ 1ˆ
0
1
1
Fubini's Theorem
¨
. If f is continuous on the rectangle R = [a, b] × [c, d], then
ˆ
dˆ b
f (x, y) dA =
R
ˆ bˆ
f (x, y) dxdy =
c
Example 3. Evaluate
a
x cos(xy) dA
R
where R = [−π/2, π/2] × [1, 5].
f (x, y) dydx
a
¨
d
c
Example 4. Find the volume of the solid S lying under the paraboloid z = x2 + y 2
and above the rectangle R = [−2, 2] × [−3, 3].
Fact
then
: If g(x) and h(y) are continuous functions of one variable and R = [a, b] × [c, d],
¨
ˆ
ˆ
b
g(x)h(y) dA =
g(x) dx
a
R
Example 5. If R = [0, ln 2] × [0, ln 5], nd
¨
e2x−y dA
R
d
h(y) dy
c