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

A child is running along a straight track in a schoolyard. The child's velocity is given by $v\left(t\right)=10e^{-0.05t}\sin \left(\frac{\pi }{48}t\right)$ for $0\le t\le 96$, where $v\left(t\right)$ is measured in meters per second, and $t$ is measured in seconds.

Find the distance between the child's position at time $t=10$ seconds and their position at time $t=70$ seconds. Show the setup for your calculations.

Find the total distance the child runs over the time interval $0\le t\le 96$ seconds. Show the setup for your calculations.

A particle, $P$, moves along the $x$-axis so that its velocity , over the interval $0\le t\le 12$, is given by the differentiable function $v_P\left(t\right)=18\sqrt[3]{t}\sin \left(0.02t^2\right)$, where $v_P\left(t\right)$ is measured in meters per second and $t$ is measured in seconds.

Find the time interval during which the velocity of particle $P$ is at least $20$ meters per second. Find the distance traveled by the particle $P$ during the time interval when the velocity of particle $P$ is at least $20$ meters per second.

At time $t=0$, particle $P$ is at position $x=-6$. A second particle $Q$, also moves along the $x$-axis such that $x_Q\left(7\right)=70$.

Using the function $v_P$ from part (a), approximate the distance between the particles $P$ and $Q$ at time $t=7$.

The density of pollen in a circular meadow ,at a distance $r$ meters from the center of the meadow, is given by an increasing, differentiable function. The pollen density is modeled by the function $g\left(r\right)=5+10e^{-0.5r}$ for $2\le r\le 8$, where $g\left(r\right)$ is measured in micrograms per square meter.

For what value of $k$, $2<k<8$, is $g\left(k\right)$ equal to the average value of $g\left(r\right)$ on the interval $2\le r\le 8$?

A particular college has a stall at a high school college fair. The college decides to give out branded pens as advertising. Students take the pens from the stall table at a rate modeled by

$f\left(t\right)=6+t\sin \left(\frac{3.5{\left(t+2\right)}^2}{5}\right)$ for $0<t\le 5$

where $f\left(t\right)$ is measured in pens per hour and $t$ is the number of hours after the start of the college fair. There are initially $30$ pens on the stall table.

After the fair has been running for two hours, the college representatives add more pens to the stall table at a rate modeled by

$g\left(t\right)=3\ln \left(2t^2-t\right)$ for $2<t\le 5$

How many pens are taken by students in the first $2$ hours of the college fair?

How many pens are on the stall table at time $t=4$?

The velocity of a particle $P$ at time $t$ is given by $v\left(t\right)=t^2-14t+45$ on the interval $0\le t\le 12$. Particle $P$ is at position $-2$ at time $t=0$.

Find the position of particle $P$ the first time it changes direction.

The density of bacteria in the petri dish, for $1\le r\le 4$, is modelled by the function $g$ defined by $g\left(r\right)=2-16\left(\text{cos}\right(1.57\sqrt{r}){)}^3$. For what value of $k$, $1<k<4$, is $g\left(k\right)$ equal to the average value of $g\left(r\right)$ on the interval $1\le r\le 4$?

From 5 A.M. to 10 A.M., the rate at which vehicles arrive at a certain toll plaza is given by $A\left(t\right)=450\sqrt{\sin (0.62t)}$, where $t$ is the number of hours after 5 A.M. and $A\left(t\right)$ is measured in vehicles per hour. Traffic is flowing smoothly at 5 A.M. with no vehicles waiting in line.

Write, but do not evaluate, an integral expression that gives the total number of vehicles that arrive at the toll plaza from 6 A.M. ($t=1$) to 10 A.M. ($t=5$).

Find the average value of the rate, in vehicles per hour, at which vehicles arrive at the toll plaza from 6 A.M. ($t=1$) to 10 A.M. ($t=5$).

An invasive species of plant appears in a fruit grove at time $t=0$ and begins to spread. The function $C$ defined by $C\left(t\right)=7.6 \text{arctan}\left(0.2t\right)$ models the number of acres in the fruit grove affected by the species $t$ weeks after the species appears. It can be shown that $C\text{'}\left(t\right)=\frac{38}{25+t^2}$.

(Note: Your calculator should be in radian mode.)

Find the average number of acres affected by the invasive species from time $t=0$ to time $t=4$ weeks. Show the setup for your calculations.

A customer at a gas station is pumping gasoline into a gas tank. The rate of flow of gasoline is modeled by a differentiable function $f$, where $f\left(t\right)$ is measured in gallons per second and $t$ is measured in seconds since pumping began. Selected values of $f\left(t\right)$ are given in the table.

Using correct units, interpret the meaning of ${\int }_{60}^{135}f\left(t\right)\text{d}t$ in the context of the problem. Use a right Riemann sum with the three subintervals $[60,90]$, $[90,120]$, and $[120,135]$ to approximate the value of ${\int }_{60}^{135}f\left(t\right)\text{d}t$.

A particle, $P$, is moving along the $x$-axis. The velocity of particle $P$ at time $t$ is given by $v_P\left(t\right)=\text{sin}\left(t^{1.5}\right)$ for $0\le t\le \pi$. At time $t=0$, particle $P$ is at position $x=5$. A second particle, $Q$, also moves along the $x$-axis. The velocity of particle $Q$ at time $t$ is given by $v_Q\left(t\right)=(t-1.8)·1.{25}^t$ for $0\le t\le \pi$. At time $t=0$, particle $Q$ is at position $x=10$.

Find the positions of particles $P$ and $Q$ at time $t=1$.

A teacher also starts reading at time $t=0$ minutes and continues reading for the next 10 minutes. The rate at which the teacher reads is modeled by the function $W$ defined by $W\left(t\right)=-\frac{3}{10}{t}^2+8t+100$, where $W\left(t\right)$ is measured in words per minute. Based on the model, how many words has the teacher read by the end of the 10 minutes? Show the work that leads to your answer.

The rate of flow of gasoline, in gallons per second, can also be modeled by $g\left(t\right)=\left(\frac{t}{500}\right)\text{cos}\left({\left(\frac{t}{120}\right)}^2\right)$ for $0\le t\le 150$. Using this model, find the average rate of flow of gasoline over the time interval $0\le t\le 150$. Show the setup for your calculations.

The area, in square feet, of the horizontal cross section at height $h$ feet is modeled by the function $f$ given by $f\left(h\right)=\frac{50.3}{e^{0.2h}+h}$.

Based on this model, find the volume of the tank. Indicate units of measure.

Fish enter a lake at a rate modeled by the function $E$ given by $E\left(t\right)=20+15\text{sin}\left(\frac{\pi t}{6}\right)$. Fish leave the lake at a rate modeled by the function $L$ given by $L\left(t\right)=4+2^{0.1t^2}$. Both $E\left(t\right)$ and $L\left(t\right)$ are measured in fish per hour, and $t$ is measured in hours since midnight ($t=0$).

(Note: Your calculator should be in radian mode.)

How many fish enter the lake over the 5-hour period from midnight ($t=0$) to 5 A.M. ($t=5$)? Give your answer to the nearest whole number.

What is the average number of fish that leave the lake per hour over the 5-hour period from midnight ($t=0$) to 5 A.M. ($t=5$)?

A tank has a height of 10 feet. The area of the horizontal cross section of the tank at height $h$ feet is given by the function $A$, where $A\left(h\right)$ is measured in square feet. The function $A$ is continuous and decreases as $h$ increases. Selected values for $A\left(h\right)$ are given in the table below.

Use a left Riemann sum with the three subintervals indicated by the data in the table to approximate the volume of the tank. Indicate units of measure.

The inside of a funnel of height 10 inches has circular cross sections, as shown in the figure above. At height $h$, the radius of the funnel is given by

$r=\frac{1}{20}(3+h^2)$

where $0\le h\le 10$. The units of $r$ and $h$ are inches.

Find the average value of the radius of the funnel.

A sports game in a stadium ends at $5\mathrm{P}.\mathrm{M}.$ and the rate at which people exit the stadium between $5\mathrm{P}.\mathrm{M}.$ and $6\mathrm{P}.\mathrm{M}.$ is given by $R\left(t\right)=350\sqrt{\sin \left(0.052t\right)}$, where $t$ is the number of minutes after $5\mathrm{P}.\mathrm{M}.$ and $R\left(t\right)$ 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 $5:15\mathrm{P}.\mathrm{M}.$ $\left(t=15\right)$to $5:45\mathrm{P}.\mathrm{M}.$ $\left(t=45\right)$.

Find the average value of the rate, in people per minute, at which people exit the stadium from $5:15\mathrm{P}.\mathrm{M}.$ $\left(t=15\right)$to $5:45\mathrm{P}.\mathrm{M}.$ $\left(t=45\right)$.

A line to exit the stadium begins to form as soon as $R\left(t\right)$ reaches 300. The number of people in line at time $t$, for $a\le t\le 50$, is given by $Q\left(t\right)={\int }_a^t\left(R\left(x\right)-300\right)\mathit{d}x$, where $a$ 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 $a\le t\le 50$. Justify your answer.

The function $f$ is defined on the closed interval [-5, 5]. The graph of $f^{\text{'}}$, the derivative of $f$, consists of two line segments and a semicircle, as shown in the figure. It is known that $f\left(4\right)=2$.

Find $f\left(0\right)$ and $f\left(-5\right)$.

For $0\le t\le 16$, a particle moves along the $x$-axis. The velocity of the particle at time $t$ is given by $v\left(t\right)=\sin \left(\frac{\pi }{8}t\right)$.

For $0\le t\le 16$, when is the particle moving to the right?

Find the total distance traveled by the particle from time $t=0$ to time $t=12$.

The particle is at position $x=-\frac{8}{\pi }$ at time $t=0$. Find the position of the particle at time $t=2$.

Water flows into a fountain at a rate modeled by the function $r$ given by

$r\left(t\right)=\{\begin{array}{ll}15\left(1-e^{-0.1t}\right)& \mathrm{for}0\le t\le 12\\ 0& \mathrm{for}t>12\end{array}$

where $r\left(t\right)$ is measured in liters per minute and $t$ is measured in minutes. Water drains from the fountain at a constant rate of $1.5$liters per minute. At time $t=0$, the fountain contains $48$ liters of water.

How much water flows into the fountain during the time interval $0\le t\le 12$?

During the time interval $0\le t\le 12$, how many liters of water are in the fountain at $t=12$?

For $t>12$, at what time $t$ does the fountain run out of water?

For $0\le t\le 12$, at what time $t$ is the amount of water in the fountain at a minimum? To the nearest liter, find the minimum volume of water in the fountain at this time. Justify your answer.

For $t\ge 0$ a particle $P$ moves along a straight line. The velocity of $P$ at time $t$ is given by $v_P\left(t\right)=5+3\sin \left(\frac{t}{2}\right)$. The particle $P$ is at position $x=-4$ at time $t=2$, where $x$ is the distance in meters and $t$ is the time in seconds.

Find the position of the particle at $t=0$.

A second particle $Q$ has position $x=11$ at $t=0$. $Q$ travels on the same straight line as $P$ at a constant velocity that is equal to the average velocity of particle $P$ in the time $0\le t\le 12$. What is the distance between particles $Q$ and $P$ at $t=2$

People enter a line for an escalator at a rate modeled by the function $r$ given by

$r\left(t\right)=\{\begin{array}{cc}44{\left(\frac{t}{100}\right)}^3{\left(1-\frac{t}{300}\right)}^7& \text{for }0\le t\le 300\\ 0& \text{for }t>300\end{array}$

where $r\left(t\right)$ is measured in people per second and $t$ is measured in seconds. As people get on the escalator, they exit the line at a constant rate of 0.7 person per second. There are 20 people in line at time $t=0$.

(Note: Your calculator should be in radian mode.)

How many people enter the line for the escalator during the time interval $0\le t\le 300$?

During the time interval $0\le t\le 300$, there are always people in line for the escalator. How many people are in line at time $t=300$?

For $t>300$, what is the first time $t$ that there are no people in line for the escalator?

A second particle, Q, also moves along the x-axis so that its velocity for $0\le t\le 4$ is given by $v_Q\left(t\right)=45\sqrt{t}·\text{cos}(0.063t^2)$ meters per hour. Find the time interval during which the velocity of particle Q is at least 60 meters per hour. Find the distance traveled by particle Q during the interval when the velocity of particle Q is at least 60 meters per hour.