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

A machine is filling a bottle of vinegar at a rate of $r\left(t\right)=12.8e^{-0.4t}$ ounces per second, where $t$ is measured in seconds. How many ounces of vinegar are put into the bottle from time $t=0$ to time $t=8$?

A particle moves along the $x$-axis. The velocity of the particle at time $t$ is $12t-3t^2$. What is the total distance traveled by the particle from time $t=0$ to $t=2$?

A particle moves along the $x$-axis. The velocity of the particle at time $t$ is given by $v\left(t\right)$, and the acceleration of the particle at time $t$ is given by $a\left(t\right)$. Which of the following gives the average velocity of the particle from time $t=0$ to time $t=12$?

A particle moves along a line so that its acceleration for $t\ge 0$ is given by $a\left(t\right)=\frac{t^2+1}{\sqrt{t^3+7}}$. If the particle's velocity at $t=0$ is $12$, what is the velocity of the particle at $t=5$?

The flow of natural gas, in cubic feet per minute, through a pipe is given by the graph shown above. Of the following, which best approximates the total number of cubic feet of natural gas that passed through the pipe during the 10 minutes shown on the graph?

A particle with velocity at any time $t$ given by $v\left(t\right)=e^{2t}$ moves in a straight line. How far does the particle move from $t=0$ to $t=4$?

What is the average value of $y$ for the part of the curve $4x-x^2$ which is in the first quadrant?

A cup of coffee, initially at a temperature of 200 degrees Fahrenheit (°F), is placed in a 60 °F room at time $t=0$ minutes. The temperature of the cup of coffee is changing at a rate of $-50e^{-0.2t}$ degrees Fahrenheit per minute. To the nearest degree, what is the temperature of the cup of coffee at time $t=3$ minutes?

On the closed interval $\left[1,5\right]$, which of the following could be the graph of a function $g$ with the property that $\frac{1}{4}{\int }_1^{5}g\left(x\right)dt=\frac{3}{2}$ ?

A particle moves along the $x$-axis so that at any time $t>0$, its acceleration is given by $a\left(t\right)=\ln \left(3^t+2\right)$. If the velocity of the particle is $-1.5$ at time $t=2$, then the velocity of the particle at time $t=4$ is

An object traveling in a straight line has position $x\left(t\right)$ at time $t$, where time is measured in seconds. If the velocity of the object is $v\left(t\right)=\sqrt[3]{2t^2+t}-5$ what is the distance traveled by the object in the first 18 seconds?

What is the average value of $y=x·\sqrt[3]{x^2-1}$ on the interval $\left[0,3\right]$?

Water is pumped into a tank at a rate of $r\left(t\right)=x\left(\frac{5}{3\sqrt[4]{x^3}}\right)$ gallons per minute, where $t$ is the number of minutes since the pump was turned on. If the tank contained $70$ gallons of water when the pump was turned on, how much water, to the nearest gallon, is in the tank after $81$ minutes?