Week 8 Tutorial Questions
Chapter 14.3
#s
2, 6, 23, 24, 26, 28, 29, 30, 34

1024
Chapter 14 / Multiple Integrals
✔
QUICK CHECK EXERCISES 14.3
(
See page 1025 for answers.
)
1.
The polar region inside the circle
r
=
2 sin
θ
and outside the
circle
r
=
1 is a simple polar region given by the inequalities
≤
r
≤
,
≤
θ
≤
2.
Let
R
betheregioninthefirstquadrantenclosedbetweenthe
circles
x
2
+
y
2
=
9 and
x
2
+
y
2
=
100. Supply the miss-
ing limits of integration.
R
f(r, θ) dA
=
✷
✷
✷
✷
f(r, θ)r dr dθ
3.
Let
V
be the volume of the solid bounded above by the
hemisphere
z
=
√
1
−
r
2
and bounded below by the disk
enclosed within the circle
r
=
sin
θ
. Expressed as a double
integral in polar coordinates,
V
=
.
4.
Express the iterated integral as a double integral in polar
coordinates.
1
1
/
√
2
x
√
1
−
x
2
1
x
2
+
y
2
dy dx
=
EXERCISE SET 14.3
1–6
Evaluate the iterated integral.
■
1.
π/
2
0
sin
θ
0
r
cos
θ dr dθ
2.
π
0
1
+
cos
θ
0
r dr dθ
3.
π/
2
0
a
sin
θ
0
r
2
dr dθ
4.
π/
6
0
cos 3
θ
0
r dr dθ
5.
π
0
1
−
sin
θ
0
r
2
cos
θ dr dθ
6.
π/
2
0
cos
θ
0
r
3
dr dθ
7–10
Use a double integral in polar coordinates to find the area
of the region described.
■
7.
The region enclosed by the cardioid
r
=
1
−
cos
θ
.
8.
The region enclosed by the rose
r
=
sin 2
θ
.
9.
The region in the first quadrant bounded by
r
=
1 and
r
=
sin 2
θ
, with
π
/
4
≤
θ
≤
π
/
2.