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(Exercises for Section 6.1: Area)
E.6.1
CHAPTER 6: APPLICATIONS OF
INTEGRALS
SECTION 6.1: AREA
Assume that distances and lengths are measured in meters.
1)
For parts a) and b) below, in the usual
xy
plane …
i)
Sketch the region
R
bounded by the graphs of the given equations.
Locate any intersection points of the graphs.
ii)
Set up the integral(s)
for the area of
R
by integrating with respect to
x
.
iii)
Set up the integral(s)
for the area of
R
by integrating with respect to
y
.
iv) Find the area of the region by using either method from ii) or iii).
ADDITIONAL PROBLEM
: Find the area using the other method.
a)
y
=
±
x
2
and
y
=
x
2
±
8
b)
y
2
=
4
±
x
and
x
+
2
y
=
1
2)
In the
tw
plane, sketch the regions bounded by the graphs of
w
=
sin
t
and
w
=
cos
t
, where
t
is restricted to the interval
0, 2
±
²
³
´
µ
, and find the total area of the
regions.
3)
Find the area of the region bounded by the graphs of the given equations in the
usual
xy
plane.
You do not have to sketch the region.
a)
y
=
3
x
2
±
5
and
y
=
x
2
+
5
x
±
2
b)
y
=
xx
2
+
16
,
x
=
0
,
x
=
3
, and
y
=
0
c)
x
±
y
3
=
0
and
x
+
y
+
2
y
2
=
0
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View Full Document(Exercises for Section 6.2: Volumes of Solids of Revolution – Disks and Washers)
E.6.2
SECTION 6.2: VOLUMES OF SOLIDS OF REVOLUTION –
DISKS AND WASHERS
Assume that distances and lengths are measured in meters.
Assume that graphs are in the usual
xy
plane, unless otherwise indicated.
1)
The region
R
is bounded by the graphs of
x
+
2
y
=
2
,
x
=
0
, and
y
=
0
.
a)
Sketch the region.
b)
Find the volume of the solid generated if
R
is revolved about the
x
axis.
c)
Find the volume of the solid generated if
R
is revolved about the
y
axis.
2)
The region
R
is bounded by the graphs of
y
=
±
x
2
and
y
=
x
2
±
8
. You should
have sketched and found the area of this region in Section 6.1, Exercise 1a.
a)
Find the volume of the solid generated if
R
is revolved about the
x
axis.
b)
Find the volume of the solid generated if
R
is revolved about the
y
axis.
3)
The region
R
in the
tw
plane is bounded by the graphs of
w
=
sin
t
,
w
=
cos
t
,
t
=
0
, and
t
=
±
4
. (This region was related to Section 6.1, Exercise 2.) Find the
volume of the solid generated if
R
is revolved about the
t
axis. Hint: You will need
to use a trig identity.
4)
The region
R
in the
xy
plane is bounded by the graphs of
y
=
cos 2
x
()
,
y
=
0
,
x
=
0
, and
x
=
4
. Sketch
R
. Find the volume of the solid generated if
R
is
revolved about the
x
axis. Hint: You will need to use a trig identity.
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 Spring '06
 Bart
 Calculus, Integrals

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