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6.3. DENSITY, MASS, AND CENTER OF MASS
6.3 Density, Mass, and Center of Mass
Motivating Questions
In this section, we strive to understand the ideas generated by the following important
questions:
• How are mass, density, and volume related?
• How is the mass of an object with varying density computed?
• What is the center of mass of an object, and how are definite integrals used to
compute it?
Introduction
We have seen in several different circumstances how studying the units on the integrand
and variable of integration enables us to better understand the meaning of a definite
integral. For instance, if v(t) is the velocity of an object moving along an axis, measured
in feet per second, while t measures time in seconds, then both the definite integral and
its Riemann sum approximation,
b
a
v(t) dt ≈
n
i=1
v(t i )△t,
have their overall units given by the product of the units of v(t) and t:
(feet/sec)·(sec) = feet.
Thus,
b
a
v(t) dt measures the total change in position (in feet) of the moving object.
This type of unit analysis will be particularly helpful to us in what follows. To begin,
in the following preview activity we consider two different definite integrals where the
integrand is a function that measures how a particular quantity is distributed over a region
and think about how the units on the integrand and the variable of integration indicate
the meaning of the integral.
Preview Activity 6.3. In each of the following scenarios, we consider the distribution of
a quantity along an axis.
(a) Suppose that the function c(x) = 200 + 100e −0.1x models the density of traffic on
a straight road, measured in cars per mile, where x is number of miles east of a
major interchange, and consider the definite integral
2
0
(200 + 100e −0.1x ) dx.
i. What are the units on the product c(x) · △x?
6.3. DENSITY, MASS, AND CENTER OF MASS
6.3 Density, Mass, and Center of Mass
Motivating Questions
In this section, we strive to understand the ideas generated by the following important
questions:
• How are mass, density, and volume related?
• How is the mass of an object with varying density computed?
• What is the center of mass of an object, and how are definite integrals used to
compute it?
Introduction
We have seen in several different circumstances how studying the units on the integrand
and variable of integration enables us to better understand the meaning of a definite
integral. For instance, if v(t) is the velocity of an object moving along an axis, measured
in feet per second, while t measures time in seconds, then both the definite integral and
its Riemann sum approximation,
b
a
v(t) dt ≈
n
i=1
v(t i )△t,
have their overall units given by the product of the units of v(t) and t:
(feet/sec)·(sec) = feet.
Thus,
b
a
v(t) dt measures the total change in position (in feet) of the moving object.
This type of unit analysis will be particularly helpful to us in what follows. To begin,
in the following preview activity we consider two different definite integrals where the
integrand is a function that measures how a particular quantity is distributed over a region
and think about how the units on the integrand and the variable of integration indicate
the meaning of the integral.
Preview Activity 6.3. In each of the following scenarios, we consider the distribution of
a quantity along an axis.
(a) Suppose that the function c(x) = 200 + 100e −0.1x models the density of traffic on
a straight road, measured in cars per mile, where x is number of miles east of a
major interchange, and consider the definite integral
2
0
(200 + 100e −0.1x ) dx.
i. What are the units on the product c(x) · △x?