(a) What is the area of a thin slice of
S
at position
x
with width d
x
?
(b) What is the mass of a small piece of
R
at position
x
with length d
x
?
(c) What is the total area of
S
?
(d) What is the total mass of
R
?
(e) What is the
x
-coordinate of the centroid of
S
?
(f) What is the centre of mass of
R
?
In Questions
8
through
10
, you will derive the formulas for the centre of mass of a rod of variable density, and
the centroid of a two-dimensional region using vertical slices (Equations 2.3.2 and 2.3.3 in the CLP–II text).
Knowing the equations by heart will allow you to answer many questions in this section; understanding
where they came from will you allow to generalize their ideas to answer even more questions.
75

A
PPLICATIONS OF
I
NTEGRATION
2.3 C
ENTRE OF
M
ASS AND
T
ORQUE
Q[8]: Suppose
R
is a straight, thin rod with density
ρ
(
x
)
at a position
x
. Let the left
endpoint of
R
lie at
x
=
a
, and the right endpoint lie at
x
=
b
.
(a) To approximate the centre of mass of
R
, imagine chopping it into
n
pieces of equal
length, and approximating the mass of each piece using the density at its midpoint.
Give your approximation for the centre of mass in sigma notation.
m
1
m
2
m
n
a
b
(b) Take the limit as
n
goes to infinity of your approximation in part (a), and express the
result using a definite integral.
Q[9]: Suppose
S
is a two-dimensional object and at (horizontal) position
x
its height is
T
(
x
)
´
B
(
x
)
. Its leftmost point is at position
x
=
a
, and its rightmost point is at position
x
=
b
.
To approximate the
x
-coordinate of the centroid of
S
, we imagine it as a straight, thin rod
R
, where the mass of
R
from
a
ď
x
ď
b
is equal to the area of
S
from
a
ď
x
ď
b
.
(a) If
S
is the sheet shown below, sketch
R
as a rod with the same horizontal length,
shaded darker when
R
is denser, and lighter when
R
is less dense.
x
y
T
(
x
)
B
(
x
)
a
b
(b) If we cut
S
into strips of very small width d
x
, what is the area of the strip at position
x
?
(c) Using your answer from (b), what is the density
ρ
(
x
)
of
R
at position
x
?
(d) Using your result from Question
8
(
b
), give the
x
-coordinate of the centroid of
S
. Your
answer will be in terms of
a
,
b
,
T
(
X
)
, and
B
(
x
)
.
Q[10]: Suppose
S
is flat sheet with uniform density, and at (horizontal) position
x
its
height is
T
(
x
)
´
B
(
x
)
. Its leftmost point is at position
x
=
a
, and its rightmost point is at
position
x
=
b
.
To approximate the
y
-coordinate of the centroid of
S
, we imagine it as a straight, thin,
vertical rod
R
. We slice
S
into thin, vertical strips, and model these as weights on
R
with:
•
position
y
on
R
, where
y
is the centre of mass of the strip, and
•
mass in
R
equal to the area of the strip in
S
.
(a) If
S
is the sheet shown below, slice it into a number of vertical pieces of equal length,
76

A
PPLICATIONS OF
I
NTEGRATION
2.3 C
ENTRE OF
M
ASS AND
T
ORQUE
approximated by rectangles. For each rectangle, mark its centre of mass. Sketch
R
as
a rod with the same vertical height, with weights corresponding to the slices you
made of
S
.