MATH 251 – LECTURE 15
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
Exercise 1. Let R = [0, 1] × [2, 3]. Compute the double integral
ZZ
1 + xy dx dy
R
Double integrals over more general domains
Definition 2. A region D is said to be of type I if it lies between the graphs of two continuous functions, that
is, if
D = {(x, y) | a ≤ x ≤ b, g1(x) ≤ y ≤ g2(x)}.
Exercise 3. Draw the domain D = {(x, y) | 0 ≤ x ≤ 1, x ≤ y ≤ 3x}.
Double integrals over more general domains
Let D = {a ≤ x ≤ b, g1(x) ≤ y ≤ g2(x)} be a domain of type I.
Choose a rectangle R = [a, b] × [c, d].
Define a new function F (x, y) by
F (x, y) =
f (x, y)
0
if (x, y) ∈ D,
if (x, y) ∈ R \ D.
Definition 4. We define the integral of f over D to be
ZZ
ZZ
F (x, y) dA,
f (x, y) dA =
D
provided that the latter integral exists.
R
Double integrals over more general domains
So, we have defined the integral of f over D. By Fubini’s theorem
ZZ
ZZ
Z b Z d
f (x, y) dA =
F (x, y) dA =
F (x, y) dy dx.
D
R
a
c
Recall that F (x, y) = 0 if y < g1(x) or if y > g2(x). Therefor,
Z d
Z g1(x)
Z g2(x)
Z
F (x, y) dy =
F (x, y) dy +
F (x, y) dy +
c
c
Z
g1 (x)
g1 (x)
Z
g2 (x)
g1 (x)
c
Z
g2 (x)
=
F (x, y) dy
g1 (x)
Z
g2 (x)
=
f (x, y) dy
g1 (x)
g2 (x)
Z
d
0 dy
F (x, y) dy +
0 dy +
=
d
g2 (x)
F (x, y) dy
Double integrals over more general domains
Exercise 5. Evaluate the double integral of f (x, y) = x over the region D = {0 ≤ x ≤ π, 0 ≤ y ≤ sin(x)}.
Double integrals over more general domains
Exercise 6. Find the volume of the solid S bounded by the surfaces z = 2x2, y = x2, and the planes z = 0 and
y = 4.
Double integrals over more general domains
Exercise 6. Find the volume of the solid S bounded by the surfaces z = 2x2, y = x2, and the planes z = 0 and
y = 4.
Double integrals over more general domains
Definition 7. A region D is said to be of type II if it lies between the graphs of two continuous functions,
that is, if
D = {(x, y) | h1(y) ≤ x ≤ h2(y), c ≤ y ≤ d}.
As before, we find that
ZZ
Z
d
Z
h2 (y)
f (x, y) dA =
D
!
f (x, y) dx dy.
c
h1 (y)
Exercise 8. Consider the type I region D = {0 ≤ x ≤ 1, 0 ≤ y ≤ x}. Express D as a type II region.
Double integrals over more general domains
2
Exercise 9. Compute the double integral of f (x, y) = ey over the type I region D = {0 ≤ x ≤ 1, 0 ≤ y ≤ x}.
Double integrals over more general domains
Exercise 10. Find the volume of the solid under the paraboloid z = 3x2 + y 2 and the region bounded by y = x
and x = y 2 − y.
Double integrals over more general domains
What if the domain D is neither of type I nor of type II?
Exercise 11. Subdivide the region {1 ≤ x2 + y 2 ≤ 4} into regions of either type I or type II.
Double integrals over more general domains
Exercise 12. Integrate the function f (x, y) = x over the region D = {1 ≤ x2 + y 2 ≤ 4}.