Lecture 11, October 17
• Double integrals. The double integral of f (x, y) over a region R in the xy-plane is
defined in terms of Riemann sums as
∫∫
n
∑
f (x∗k , yk∗ ) ∆Ak .
f (x, y) dA = lim
n→∞
R
k=1
If f is positive, then this expression gives the volume of the solid that lies below the
graph of f and above the region R in the xy-plane. When R = [a, b] × [c, d], one has
∫∫
∫ d∫ b
∫ b∫ d
f (x, y) dA =
f (x, y) dx dy =
f (x, y) dy dx.
c
R
a
a
c
.....................................................................................
Example 1. Consider f (x, y) = x2 y over the rectangle R = [0, 2] × [0, 1]. Then
∫∫
∫ 1∫ 2
f (x, y) dA =
x2 y dx dy
0
R
0
and we can focus on the inner integral first. Integrating with respect to x, we get
[ 3 ]2
∫ 2
∫ 2
8y
x
2
2
=
x y dx = y
x dx = y
3 x=0
3
0
0
and so the double integral is
∫∫
∫
f (x, y) dA =
0
R
1
[ 2 ]1
4y
4
8y
= .
dy =
3
3 y=0 3
Alternatively, one may reach the same answer by writing
∫∫
∫ 2∫ 1
f (x, y) dA =
x2 y dy dx
R
0
0
and by integrating with respect to y first. This approach gives
[ 2 ]1
∫ 1
∫ 1
x2
2
2
2 y
x y dy = x
y dy = x
=
2 y=0
2
0
0
for the inner integral, so the double integral is equal to
[ 3 ]2
∫∫
∫ 2 2
x
x
4
f (x, y) dA =
dx =
= .
6 x=0 3
0 2
R