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Math 251-copyright Joe Kahlig, 15A
Section 13.2: Iterated Integrals
Definition: Suppose f is a function of two variables that is integrable over the rectangle
R = [a, b] × [c, d].
Zb
The partial integration of f with respect to x is
f (x, y)dx and means that y is held fixed and f
a
is integrated with respect to x.
Zd
The partial integration of f with respect to y is
f (x, y)dy and means that x is held fixed and f
c
is integrated with respect to y.
Both of these integrals give an answer that is a function of the variable that is held fixed. Integrating
this function with respect to the variable held fixed is called an iterated integral.




Zb Zd
Zd Zb
 f (x, y)dy  dx
 f (x, y)dx dy
a
c
c
Example: Evaluate the following.
Z3 Z2
(A)
0
1
Z2 Z3
(B)
1
6x2 y dydx
0
6x2 y dxdy
a
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Math 251-copyright Joe Kahlig, 15A
Fubini’s Theorem: If f is continuous on the rectangle R = [a, b] × [c, d], then
Zb Zd
ZZ
f (x, y)dA =
f (x, y)dydx =
a
R
Zd Zb
c
f (x, y)dxdy
c
a
In the case where f (x, y) = g(x)h(y) then
Zb Zd
ZZ
f (x, y)dA =
R
Zb
g(x)h(y)dydx =
a
c
Zd
g(x)dx
a
h(y)dy
c
Example: Evaluate the double integral where R = {(x, y)|0 ≤ x ≤ 3, 1 ≤ y ≤ 5} = [0, 3] × [1, 5]
ZZ
(x + 3y 2 )dA
R
Example: If R = [1, 8] × [0, π], evaluate
ZZ
y cos(xy)dA
R
Math 251-copyright Joe Kahlig, 15A
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Example: find the volume of the solid S that is bounded by the elliptic paraboloid z = 6x2 + y 2 + 1,
the planes x = 3 and y = 4 and the three coordinate planes.