AP Calculus – Integral Applications
t2 t3
 , for
2 60
0 ≤ t ≤ 30 minutes. R(t) is measured in people per hour. There are no people in the theatre at t = 0 when the
doors open and the show begins at t = 30.
a. How many people are in the theatre when the show begins?
b. Find the time when the rate at which people enter the theatre is increasing the fastest. Express this rate
using the proper units. Justify your answer.
c. The total wait time for all the people in the theatre is found by adding the time each person waits for the
show to begin, starting at the time the person enters the theatre. The function w models the total wait time
for all the people who enter the theatre before time t. The derivative of w is given by w′(t)=(30 – t)R(t).
Find the total wait time for people who enter the theatre after time t = 0.
d. On average, how long does each person who sees the show wait in the theatre for the show to begin?
1. The rate that people enter a Broadway theatre is modeled by the function R given by R(t ) 
2. (1988AB2-no calculator)
A particle moves along the x-axis so that its velocity at any time t ≥ 0 is given by v(t) = 1 – sin(2πt)
a. Find the acceleration a(t) of the particle at any time t.
b. Find all value of t, 0 ≤ t ≤ 2, for which the particle is at rest.
c. Find the position x(t) of the particle at any time t if x(0)=0
3. (1989AB3-no calculator) A particle moves along the x-axis in such a way that its acceleration at time t
for t ≥ 0 is given by a(t) = 4cost(2t). At time t = 0, the velocity of the particle is v(0) = 1 and its position is
x(0) = 0.
a. Write an equation for the velocity v(t) of the particle.
b. Write an equation for the position x(t) of the particle.
c. For what value of t, 0 ≤ t ≤ π, is the particle at rest?