4. Double Integrals
(from Stewart, Calculus, Chapter 15)
Partial Integrals:
R
d
c
f(x, y) dy is calculated by holding x
constant and integrating with respect t o y from y = c to y = d.
Note th at th e resul t is a function of x.
For each fixed x, the t rac e of f(x, y)isacurveC in the plane
x = constant. The partial integral A(x)=
R
d
c
f(x, y) dy is the
area und er th e curve C from y = c to y = d.
Similarly,
R
b
a
f(x, y) dx is calculate d by holding y constant and integrating with respect to x from x = a to
x = b. The result is a function of y.
Finding Volume with a Dou b le Integral:
RR
D
f(x, y)dA is the signed volume between the surface
z = f (x, y) and the region D in the xy-plane.
Finding Area with a Dou bl e Integral:
Area(D)=
RR
D
1 dA
Iterated Integrals
Vertically Simple Regions:
D = {(x, y):a x b, g
1
(x) y g
2
(x)}
ZZ
D
f(x, y) dA =
Z
b
a
Z
g
2
(x)
g
1
(x)
f(x, y) dy dx
Horizontally Simple Regions:
D = {(x, y):c y d, h
1
(y) x h
2
(y)}
ZZ
D
f(x, y) dA =
Z
d
c
Z
h
2
(y )
h
1
(y )
f(x, y) dx dy
Ex. 1. Find the volume of the solid that lies u nd er the surface z = xy and above the region D in the
xy-plane bounded by the line y = x 1 and the parabola y
2
=2x + 6.
Ex. 2. Evaluate
Z
1
0
Z
1
x
sin(y
2
) dy dx.
Ex. 3. Reverse the order of integration in the integral
Z
9
4
Z
p
y
2
f(x, y) dx dy.
Ex. 4. Evaluate
ZZ
D
(x +2y) dA,whereD is the region bounded by the parabolas y =2x
2
and y =1+x
2
.
Polar Coordinates (r, ✓)
r is th e signe d dist anc e from the origin
✓ is the angle measured counter-clo ckwise from the
positive x-axi s
x = r cos(✓) y = r sin(✓)
r
2
= x
2
+ y
2
tan(✓)=
y
x
Double Integrals in Polar Coordinates:
ZZ
R
f(x, y) dA =
Z
↵
Z
b
a
f(r cos ✓, r sin ✓) rdrd✓.
Ex. 5. Evaluate
ZZ
R
(3x +4y
2
)dA where R is the region in the upper half plane x 0 bounded by the
circles x
2
+ y
2
= 1 and x
2
+ y
2
= 4.
Ex. 6. Evaluate the integral
Z
0
1
Z
p
1x
2
0
cos(x
2
+ y
2
) dy dx.
Ex. 7. Find the volume of the region bounded by the paraboloids z = x
2
+ y
2
and z =8 x
2
y
2
.
Ex. 8. Evaluate the integrals.
(a)
Z
1
0
Z
1
x
2
x
3
sin(y
3
) dy dx
(b)
ZZ
D
e
x
2
dA where D is the region bounded by the lines y =2x, y = 0, and x = 1.
(c)
ZZ
D
e
x
2
+y
2
dA where D is the top half of the disk centered at the origin of radius 3.
