MATH 251 – LECTURE 14
JENS FORSGÅRD
http://www.math.tamu.edu/~jensf/
This week: 13.1–4
webAssign: 13.1–3, due 2/29 11:55 p.m.
Next week: 13.4–6,8
webAssign: 13.4-6, opens 2/29 12 a.m.
Help Sessions:
M W 5.30–8 p.m. in BLOC 161
Office Hours:
BLOC 641C
M 12:30–2:30 p.m.
W 2–3 p.m.
or by appointment.
Double integrals over rectangles
Consider a rectangle
R = [a, b] × [c, d] = {(x, y) | a ≤ x ≤ b, c ≤ y ≤ d}.
We want to define the integral of a function f (x, y) over the rectangle R.
Choose a partitioning P of the intervals [a, b] and [c, d] by
a = x0 < x1 < · · · < xm−1 < xm = b
Let
∆xi = xi − xi−1
Let Rij be the rectangle
and
and
c = y0 < y1 < · · · < yn−1 < yn = d.
∆yj = yj − yj−1.
Rij = [xi−1, xi] × [yj−1, yj ],
so that Rij has area ∆Aij = ∆xi∆yj .
Double integrals over rectangles
For each rectangle Rij , choose a point (x∗ij , yij∗ ) contained in this rectangle. The Riemann sum of f (x, y)
(corresponding to the above partition and choices of points) is
n
m X
X
f (x∗ij , yij∗ )∆Aij .
i=1 j=1
Let |P | = maxij
p
(∆xi)2 + (∆yi)2.
Definition 1. The double integral of f over the rectangle R is
ZZ
m X
n
X
f (x∗ij , yij∗ )∆Aij ,
f (x, y) dA = lim
|P |→0
R
i=1 j=1
provided that the limit exists.
Since ∆Aij = ∆xi∆yj , we usually write
ZZ
ZZ
f (x, y) dA =
R
f (x, y) dx dy
R
Approximations with Riemann sums
If |P | is small, then
ZZ
f (x, y) dx dy ≈
R
m X
n
X
f (x∗ij , yij∗ )∆Aij .
i=1 j=1
Exercise 2. Approximate the integral of f (x, y) = x − y 2 over the rectangle R = [0, 2] × [1, 3] by partition
the sides of the rectangle into two parts of equal lengths, and by taking (x∗ij , yij∗ ) to be the midpoint of each
rectangle in this partition.
Volumes
Double integrals can be interpreted as volumes (computed with signs).
Let f (x, y) be non-negative on the rectangle R. The graph of f is a surface with equation
z = f (x, y).
Let S be the solid between R and the graph of f , i.e.,
S = {(x, y, z) | (x, y) ∈ R, 0 ≤ z ≤ f (x, y)},
and let V = Vol(S). Then,
ZZ
ZZ
V =
f (x, y) dA =
R
f (x, y) dx dy.
R
Volumes
Exercise 3. Compute the double integral of f (x, y) = 1 + x over the rectangle R = [0, 1] × [0, 1].
Iterated integrals
Theorem 4 (Fubini’s Theorem). If f is continuous on the rectangle R = [a, b] × [c, d], then
Z b Z d
Z d Z b
ZZ
f (x, y) dy dx
f (x, y) dx dy =
f (x, y) dx dy =
R
c
a
a
c
Exercise 5. Compute the double integral of f (x, y) = 1 + x over the rectangle R = [0, 1] × [0, 1].
Iterated integrals
Exercise 6. Compute the iterated integral
Z 1Z
1
4
0
0
√
x + y dx dy.