AP Calc Notes: DI – 9 Accumulation and Net Change
f (b) − f ( a )
;
Arrange the following: f ' ( a ) ;
b−a
b
∫
a
b
1
f ( x )dx ;
f ( x )dx
b − a ∫a
Units
Formula
Accumulation from x = a to x = b
Average value on [a, b]
Average rate of change on [a, b]
Rate of change at x = a
1. Suppose C(t) represents the daily cost of heating your house, measured in dollars per day, where t is
time measured in days and t = 0 corresponds to January 1, 2000.
90
a. Interpret
∫ C (t )dt .
0
b. Write an expression that represents the average cost per day to heat your house from January 1, 2000
to February 1, 2000.
2. Let P(t ) = 67.38(1.026)t . P(t) represents the population of Mexico where P is in millions of people and
t is in years since 1980.
a. Find the average population of Mexico from 1980 to 2000.
b. Predict the average population of Mexico from 2000 to 2020.
c. What are the units of
∫ P(t )dt and what could this measure?
3) A far north oil pipeline develops a leak which is not discovered until oil has covered 6000 square yards of
the frozen ground. After the leak is discovered, the area of the oil spill continues to spread at a rate of
r(t) = 1296 + 72t – 3t2 square yards per hour, where t is hours after the leak is discovered.
a. How much additional ground is covered in oil 6 hours after the discovery?
ΔA =
∫
6
0
r(t)dt = 8856 square yards
b. How much total ground is covered in oil 36 hours after the discovery?
A(36) = A(0) +
c. What does
∫
36
0
∫ r ( t ) dt
24
r(t)dt = 6000 + 46656 = 52656 square yards
represent?
6
The change in area of ground covered by oil from t = 6 hours to t = 24 hours after
discovery of the spill.
d. What does
24
1
r ( t ) dt represent?
∫
24 − 6 6
The average rate of change in area of ground covered by oil from t = 6 hours to t = 24
hours after discovery of the spill; the average number of addition square yards covered
each hour.