Math 217: Multiple Integrals (Ch. 15)
Lecture 15 (Oct. 16-17)
Double Integrals over Rectangles (reading: 15.1)
Recall the definition of the Riemann integral for a function of a single variable:
For a function of 2 variables, we define the double integral over a rectangle
R = [a, b] ⇥ [c, d] analogously:
Definition:
ZZ
f (x, y)dA := lim
R
n,m!1
m X
n
X
f (x⇤ij , yij⇤ ) x y
i=1 j=1
(if this limit exists).
Theorem: if f (x, y) is continuous on R, then this limit exists (and is independent of
the choices of (x⇤ij , yij⇤ )).
Proof: omitted.
1
Definition: Let f (x, y) be a continuous, non-negative function on a rectangle R. Then
the volume under the graph of f is
V :=
ZZ
f (x, y)dA.
R
Remark on notation:
ZZ
f (x, y)dA =
R
ZZ
p
RR
Example: approximate [0,1]⇥[0,1] 1
then find the exact value.
f (x, y)dxdy =
R
ZZ
f dA.
R
y 2 dA by a Riemann sum with m = n = 2,
Midpoint rule for approximating double integrals:
ZZ
R
f (x, y)dA ⇡
where x̄i := (xi 1 + xi )/2, ȳj := (yj
(This is what we just used!).
1
m X
n
X
i=1 j=1
+ yj )/2.
2
f (x̄i , ȳj ) x y
Average value
Recall the average value of a function in one-variable calculus:
Definition: The average value of a function f (x, y) over the rectangle R is
fav
1
:=
area(R)
Example: the average value of f (x, y) =
ZZ
f (x, y)dA.
R
p
1
y 2 on [0, 1]2 is:
Properties of double integrals:
•
•
RR
RR
(f (x, y) + g(x, y))dA =
(cf (x, y))dA = c
• f (x, y)
g(x, y)
RR
RR
f (x, y)dA +
RR
g(x, y)dA
f (x, y)dA
=)
RR
f (x, y)dA
RR
g(x, y)dA.
The fact that these statements hold for integrals follows from the fact that they hold
for the approximating Riemann sums. We do not give a proof, though.
3
Iterated Integrals (reading: 15.2)
Question: how do we actually compute
RR
R
f dA? (Not by the definition!)
Answer: by performing an iterated integral:
Theorem: (Fubini’s Theorem). Suppose f (x, y) is continuous on R := [a, b] ⇥ [c, d].
Then
ZZ
f (x, y)dxdy =
R
Z
d
c
Z
b
f (x, y)dx dy =
a
Z b Z
a
4
d
f (x, y)dy dx.
c
Example: Compute
RR
[0,1]2
p
1
y 2 dA.
Example: Sometimes one order is much easier than the other: compute
5
RR
[1,2]⇥[0,⇡]
y sin(xy)dxdy.