Lecture 12, October 19
• Fubini’s theorem. If f is continuous over a bounded region R, then
f (x, y) dA =
f (x, y) dy dx =
f (x, y) dx dy
R
for some suitable limits of integration that describe the region R. When it comes to
the middle integral, one needs to find the possible values of y for each fixed value of x
and that corresponds to a description of R using vertical slices. When it comes to the
rightmost integral, one uses horizontal slices, instead.
.....................................................................................
Example 1. Switching the order of integration, one finds that
2 1
1 y
1
1 1
ey
e−1
2
2
2
ey dy dx =
ey dx dy =
yey dy =
=
.
2
2
0
x
0
0
0
0
Example 2. We switch the order of integration in order to compute the integral
4 2
2 2x
cos(2y/x)
cos(2y/x)
I=
dx dy =
dy dx.
x
x
0
y/2
0
0
In this case, the inner integral is given by
y=2x
1 2x
1 x sin(2y/x)
sin 4
cos(2y/x) dy =
=
x 0
x
2
2
y=0
and so the double integral is equal to
2 2x
2
cos(2y/x)
sin 4
I=
dy dx =
dx = sin 4.
x
2
0
0
0
4
y = 2x
1
y=x
1
2
Figure: The regions of integration for Examples 1 and 2.