CALCULUS
WORKSHEET ON APPLICATIONS OF THE DEFINITE INTEGRAL - ACCUMULATION
Work the following on notebook paper.
1. A tire factory is located at the center of a town, which extends three miles either side of the
factory. Let x be the distance from the factory, where x is measured in miles. A straight
highway goes through town, and the density of particles of pollutant along the highway is
given by p  x   500 12  x 2 , where p  x  is the number of particles of pollutant per mile


per day at the point x. Find the total number of particles of pollutant per day deposited in
the town along the highway.
2. The density of cars (in cars per mile) down a 10-mile stretch of the Massachusetts Turnpike
starting at a toll plaza is given by p  x   400  100sin  x  where x is the distance in
miles from the toll plaza and 0  x  10 .
(a) Write a Riemann sum that approximates the total number of cars down the 10-mile
stretch.
(b) Find the total number of cars on the 10-mile stretch by evaluating a definite integral.
3. A city in the shape of a circle of radius 10 miles is growing
with its population density a function of the distance from the
center of the city. At a distance of r miles from the city center,
5000
its population density is P  r  
people/square mile. In
1 r
the figure, thin rings have been drawn around the center of the city.
The area in square miles of the shaded ring can be approximated by
the product 2 r  r , where r is the radius of the outer ring.
(a) Write an expression to represent the number of people in the
shaded ring.
(b) Write a Riemann sum that approximate the population of the city.
(c) Find the exact population of the city by evaluating a definite integral.
4. Greater Seattle can be approximated by a semicircle of radius 5 miles with its center on
the shoreline of Puget Sound. Moving away from the center along a radius, the population
density can be approximated by p  x   20,000e 0.13x people per square mile.
(a) Write a Riemann sum that approximates the population of the city.
(b) Find the exact population of the city by evaluating a definite integral.
5. The density of oil in a circular oil slick on the surface of the ocean at a distance r meters
50
kg/m 2 .
from the center of the slick is given by f  r  
1 r
(a) If the slick extends from r = 0 to r = 10,000m, find a Riemann sum approximating the
total mass of oil in the slick.
(b) Find the exact value of the mass of oil in the slick by turning your sum into an integral
and evaluating it.
(c) Within what distance r is half the oil of the slick contained?
Answers to Worksheet on Applications of the Definite Integral
1.
2
  3 500 12  x  dx  27,000 particles2.
3
n
2) (a)
 p  xi   xi (b)  0
i 1
10
400  100sin  x  dx  4000 people
 5000 
 10, 000 r 
3. (a) 
  2 r  r   
  r 
 1 r 
 1 r 
10 10, 000 r
dr = 238,827 people
(c) 
0
1 r
  xi    20, 000e 0.13x    xi 
n
4. (a)
i
i 1
n
n
 50
p
x


x



 i i  2 xi  1  x
i

i 1
i 1
k
 50 
(c)  2 x 
 dx  = 1,,569,349
0
 1 x 
5. (a)
(b)
 10, 000 ri 
   ri 

i 1  1  ri
n
(b)

 0  x  20,000e

   xi  (b)

5
10000
0
 0.13 x
 dx  515,387 people
 50 
2 x 
 dx = 3,138,699 kg of oil
1 x 
k = 5003.91 meters