6.3. DENSITY, MASS, AND CENTER OF MASS
355
ii. What are the units on the definite integral and its Riemann sum approximation given by
2
0
c(x) dx ≈
n
i=1
c(x i )△x?
iii. Evaluate the definite integral
2
0
c(x) dx =
2
0
(200 + 100e −0.1x ) dx and write
one sentence to explain the meaning of the value you find.
(b) On a 6 foot long shelf filled with books, the function B models the distribution
of the weight of the books, measured in pounds per inch, where x is the number
of inches from the left end of the bookshelf. Let B(x) be given by the rule
B(x) = 0.5 +
1
(x+1) 2 .
i. What are the units on the product B(x) · △x?
ii. What are the units on the definite integral and its Riemann sum approximation given by
36
12
B(x) dx ≈
n
i=1
B(x i )△x?
iii. Evaluate the definite integral
72
0
B(x) dx =
72
0
(0.5 +
1
(x+1) 2 ) dx and write
one sentence to explain the meaning of the value you find.
⊲⊳
Density
The mass of a quantity, typically measured in metric units such as grams or kilograms, is
a measure of the amount of a quantity. In a corresponding way, the density of an object
measures the distribution of mass per unit volume. For instance, if a brick has mass 3 kg
and volume 0.002 m 3 , then the density of the brick is
3kg
0.002m 3 = 1500
kg
m 3 .
As another example, the mass density of water is 1000 kg/m 3 . Each of these relationships
demonstrate the following general principle.
For an object of constant density d, with mass m and volume V ,
d =
m
V
, or m = d · V .
But what happens when the density is not constant?
355
ii. What are the units on the definite integral and its Riemann sum approximation given by
2
0
c(x) dx ≈
n
i=1
c(x i )△x?
iii. Evaluate the definite integral
2
0
c(x) dx =
2
0
(200 + 100e −0.1x ) dx and write
one sentence to explain the meaning of the value you find.
(b) On a 6 foot long shelf filled with books, the function B models the distribution
of the weight of the books, measured in pounds per inch, where x is the number
of inches from the left end of the bookshelf. Let B(x) be given by the rule
B(x) = 0.5 +
1
(x+1) 2 .
i. What are the units on the product B(x) · △x?
ii. What are the units on the definite integral and its Riemann sum approximation given by
36
12
B(x) dx ≈
n
i=1
B(x i )△x?
iii. Evaluate the definite integral
72
0
B(x) dx =
72
0
(0.5 +
1
(x+1) 2 ) dx and write
one sentence to explain the meaning of the value you find.
⊲⊳
Density
The mass of a quantity, typically measured in metric units such as grams or kilograms, is
a measure of the amount of a quantity. In a corresponding way, the density of an object
measures the distribution of mass per unit volume. For instance, if a brick has mass 3 kg
and volume 0.002 m 3 , then the density of the brick is
3kg
0.002m 3 = 1500
kg
m 3 .
As another example, the mass density of water is 1000 kg/m 3 . Each of these relationships
demonstrate the following general principle.
For an object of constant density d, with mass m and volume V ,
d =
m
V
, or m = d · V .
But what happens when the density is not constant?