Definite Integrals in Context (College Board AP® Calculus AB)

Particle moves along the -axis such that, for time , its velocity is given by . At time , the position of particle is .

Find , the position of particle at time .

A child is running along a straight track in a schoolyard. The child's velocity is given by for , where is measured in meters per second, and is measured in seconds.

Find the distance between the child's position at time seconds and their position at time seconds. Show the setup for your calculations.

Find the total distance the child runs over the time interval seconds. Show the setup for your calculations.

A sports game in a stadium ends at 5 PM and the rate at which people exit the stadium between 5 PM and 6 PM is given by , where is the number of minutes after 5 PM and is measured in people per minute.

Write, but do not evaluate, an integral expression that gives the total number of people that exit the stadium from 5:15 PM to 5:45 PM .

Find the average value of the rate, in people per minute, at which people exit the stadium from 5:15 PM to 5:45 PM .

A line to exit the stadium begins to form as soon as reaches 300. The number of people in line at time , for , is given by , where is the time when a line first begins to form. To the nearest whole number, find the greatest number of people in line to exit the stadium in the time interval . Justify your answer.

The function is defined on the closed interval [-5, 5]. The graph of , the derivative of , consists of two line segments and a semicircle, as shown in the figure. It is known that .

Find and .