358
6.3. DENSITY, MASS, AND CENTER OF MASS
distributed according to the density function ρ(x) = 2e −0.2x , where x is the
distance in cm from the left end of the rod, and the units on ρ(x) are g/cm. If
the rod is 10 cm long, determine the exact mass of the rod.
(b) Consider the cone that has a base of radius 4 m and a height of 5 m. Picture
the cone lying horizontally with the center of its base at the origin and think of
the cone as a solid of revolution.
i. Write and evaluate a definite integral whose value is the volume of the
cone.
ii. Next, suppose that the cone has uniform density of 800 kg/m 3 . What is
the mass of the solid cone?
iii. Now suppose that the cone’s density is not uniform, but rather that the
cone is most dense at its base. In particular, assume that the density of the
cone is uniform across cross sections parallel to its base, but that in each
such cross section that is a distance x units from the origin, the density of
the cross section is given by the function ρ(x) = 400 +
200
1+x 2 , measured in
kg/m 3 . Determine and evaluate a definite integral whose value is the mass
of this cone of non-uniform density. Do so by first thinking about the mass
of a given slice of the cone x units away from the base; remember that in
such a slice, the density will be essentially constant.
(c) Let a thin rod of constant cross-sectional area 1 cm 2 and length 12 cm have
its mass be distributed according to the density function ρ(x) =
1
25 (x − 15) 2 ,
measured in g/cm. Find the exact location z at which to cut the bar so that the
two pieces will each have identical mass.
⊳
Weighted Averages
class
grade grade points credits
chemistry
B+
3.3
5
calculus
A3.7
4
history
B2.7
3
psychology
B2.7
3
Table 6.1: A college student’s semester grades.
The concept of an average is a natural one, and one that we have used repeatedly as
part of our understanding of the meaning of the definite integral. If we have n values a 1 ,
a 2 , . . ., a n , we know that their average is given by
a 1 + a 2 + · · · + a n
n
,
6.3. DENSITY, MASS, AND CENTER OF MASS
distributed according to the density function ρ(x) = 2e −0.2x , where x is the
distance in cm from the left end of the rod, and the units on ρ(x) are g/cm. If
the rod is 10 cm long, determine the exact mass of the rod.
(b) Consider the cone that has a base of radius 4 m and a height of 5 m. Picture
the cone lying horizontally with the center of its base at the origin and think of
the cone as a solid of revolution.
i. Write and evaluate a definite integral whose value is the volume of the
cone.
ii. Next, suppose that the cone has uniform density of 800 kg/m 3 . What is
the mass of the solid cone?
iii. Now suppose that the cone’s density is not uniform, but rather that the
cone is most dense at its base. In particular, assume that the density of the
cone is uniform across cross sections parallel to its base, but that in each
such cross section that is a distance x units from the origin, the density of
the cross section is given by the function ρ(x) = 400 +
200
1+x 2 , measured in
kg/m 3 . Determine and evaluate a definite integral whose value is the mass
of this cone of non-uniform density. Do so by first thinking about the mass
of a given slice of the cone x units away from the base; remember that in
such a slice, the density will be essentially constant.
(c) Let a thin rod of constant cross-sectional area 1 cm 2 and length 12 cm have
its mass be distributed according to the density function ρ(x) =
1
25 (x − 15) 2 ,
measured in g/cm. Find the exact location z at which to cut the bar so that the
two pieces will each have identical mass.
⊳
Weighted Averages
class
grade grade points credits
chemistry
B+
3.3
5
calculus
A3.7
4
history
B2.7
3
psychology
B2.7
3
Table 6.1: A college student’s semester grades.
The concept of an average is a natural one, and one that we have used repeatedly as
part of our understanding of the meaning of the definite integral. If we have n values a 1 ,
a 2 , . . ., a n , we know that their average is given by
a 1 + a 2 + · · · + a n
n
,