Calculus 2nd Edition Rogawski

SECTION 6.2 Setting Up Integrals: Volume, Density, Average Value 307
In general, density is a function ρ(x, y)
that depends not just on the distance to
the origin but also on the coordinates
(x, y). Total mass or population is then
computed using double integration, a topic
in multivariable calculus.
Ring of
width r,
Area ≈ 2πr i r
r 1 r 2 r i R
FIGURE 8 Dividing the circle of radius R
into N thin rings of width r = R/N.
Remember that for a radial density
function, the total population is obtained
by integrating 2πrρ(r) rather than ρ(r).
In some situations, density is a function of distance to the origin. For example, in
the study of urban populations, it might be assumed that the population density ρ(r) (in
people per square kilometer) depends only on the distance r from the center of a city. Such
a density function is called a radial density function.
We now derive a formula for the total population P within a radius R of the city
center assuming a radial density ρ(r). First, divide the circle of radius R into N thin rings
of equal width r = R/N as in Figure 8.
N∑
Let P i be the population within the ith ring, so that P = P i . If the outer radius of
the ith ring is r i , then the circumference is 2πr i , and if r is small, the area of this ring
is approximately 2πr i r (outer circumference times width). Furthermore, the population
density within the thin ring is nearly constant with value ρ(r i ). With these approximations,
i=1
P i ≈ 2πr i r ×
} {{ }
ρ(r i )
} {{ }
= 2πr i ρ(r i )r
Area of ring Population
density
P =
N∑
N∑
P i ≈ 2π r i ρ(r i )r
This last sum is a right-endpoint approximation to the integral 2π
i=1
i=1
∫ R
0
rρ(r) dr.As N tends
to ∞, the approximation improves in accuracy and the sum converges to the integral. Thus,
for a population with a radial density function ρ(r),
Population P within a radius R = 2π
∫ R
0
rρ(r) dr 4
EXAMPLE 5 Computing Total Population The population in a certain city has radial
density function ρ(r) = 15(1 + r 2 ) −1/2 , where r is the distance from the city center in
kilometers and ρ has units of thousands per square kilometer. How many people live in
the ring between 10 and 30 km from the city center?
Solution The population P (in thousands) within the ring is
P = 2π
∫ 30
10
r ( 15(1 + r 2 ) −1/2) dr = 2π(15)
∫ 30
10
r
(1 + r 2 dr
) 1/2
Now use the substitution u = 1 + r 2 , du = 2rdr. The limits of integration become
u(10) = 101 and u(30) = 901:
∫ 901
( ) 1
P = 30π u −1/2 du = 30πu 1/2∣ ∣901
≈ 1881 thousand
2
101
101
In other words, the population is approximately 1.9 million people.
Flow Rate
When fluid flows through a tube, the flow rate Q is the volume per unit time of fluid
passing through the tube (Figure 9). The flow rate depends on the velocity of the fluid particles.
If all particles of the fluid travel with the same velocity v (say, in units of cm 3 /min),
and the tube has radius R, then
Flow rate Q = cross-sectional area × velocity = πR
} {{ }
2 v cm 3 /min
Volume per unit time