A pump fills a vessel at rate $P(t)$, in L/hr. Use the trapezoidal rule on all subintervals in the table to estimate ${\int }_0^{5}P(t)dt$.

A drone rises at rate $R(t)$, in m/min. Using the trapezoidal rule on the table's subintervals, estimate ${\int }_0^{4}R(t)dt$.

A machine produces material at rate $M(t)$, in kg/hr. Use a trapezoidal approximation over every interval in the table to estimate ${\int }_0^{4}M(t)dt$.

A robot travels at rate $v(t)$, in m/min. Based on the table, use the trapezoidal rule to estimate ${\int }_0^{6}v(t)dt$.

Water enters a container at rate $W(t)$, in L/min. Use trapezoids on the table's three subintervals to estimate ${\int }_0^{5}W(t)dt$.

Snow falls at rate $S(t)$, in cm/hr. Use the trapezoidal rule on the subintervals shown to estimate ${\int }_0^{7}S(t)dt$.

The rate $E(t)$ measures carbon dioxide emitted in kg/hr. What does ${\int }_0^{8}E(t)dt$ represent?

A stream's flow rate is $Q(t)$ in L/min. Interpret $\frac{1}{10}{\int }_0^{10}Q(t)dt$.

A bacterium population changes at rate $B(t)$ in cells/day. What are the units of ${\int }_2^{6}B(t)dt$?

The temperature-change rate of a sample is $T^{\prime }(t)$ in degrees Celsius per minute. What does ${\int }_1^5{T}^{\prime }(t)dt$ give?

A cyclist's velocity is $v(t)$ in km/hr. If $v(t)\ge 0$ for $0\le t\le 3$, what does ${\int }_0^{3}v(t)dt$ represent?

The graph shows $R(t)$, the rate cars enter a garage in cars/hr. How many cars enter from $t=0$ to $t=8$ hours?

The graph shows $F(t)$, an inflow rate in cubic meters per minute. What volume enters from $t=0$ to $t=5$ minutes?

The graph gives an algae colony's growth rate $G(t)$ in g/day. Find the colony's mass increase from day 0 to day 3.

The graph gives a runner's velocity $v(t)$ in km/hr. Because the graph is above the axis, what distance does the runner travel from hour 0 to hour 4?

The graph shows medicine entering a patient at rate $I(t)$ in mL/min. How much medicine enters during the first 6 minutes?

The graph shows $A(t)$, an algae colony's growth rate in g/day. What is the net mass increase from day 0 to day 4?

Let $g(x)={\int }_0^{x}f(t)dt$, where $f$ is continuous. A student says $g$ has a relative maximum at $x=2$ because $f(2)=0$. Which response is valid?

A tank has volume $V(t)=100+{\int }_0^{t}q(s)ds$, where $q$ is continuous in L/min. A student claims $V$ increases on $[0,5]$ because $q(1)>0$. Which response is valid?

Let $H(x)={\int }_2^x(t^2+1)dt$. A student states that $H^{\prime }(3)=10$ by the Fundamental Theorem of Calculus. Is the justification valid?

Let $A(x)={\int }_0^{x}f(t)dt$. A student claims $A$ is concave up at $x=4$ because $f(4)>0$. Which response is valid?

Rain falls at a continuous rate $r(t)\ge 0$ in mm/hr. A student says ${\int }_0^{12}r(t)dt$ is the total rainfall during 12 hours. Is the justification valid?

A robot's velocity is $v(t)=3t^2-12t+9$ m/min for $0\le t\le 3$. What is its total distance traveled, in meters?

Liquid enters a container at rate $q(t)=5e^{0.2t}$ L/min. How many liters enter during the first 2 minutes? Round to three decimals.

A sample's temperature changes at rate $r(t)=4-t$ degrees Celsius per minute for $0\le t\le 5$. What is the net temperature change?

Water enters or leaves a tank at rate $q(t)=6-2t$ L/min for $0\le t\le 5$. What is the tank's net volume change over this interval?

A reservoir has volume $V(x)=500+{\int }_0^{x}W(t)dt$ cubic meters, where $W$ is in cubic meters/day. What does $V^{\prime }(3)$ represent?

The amount of dye in a tank is $D(t)=40+{\int }_0^{t}c(s)ds$ mg, where $c$ is in mg/min. What does $D^{\prime }(5)$ represent?

A compost pile's mass is $M(x)=12+{\int }_2^{x}k(t)dt$ kg, where $k$ is in kg/day. Which expression equals $M^{\prime }(x)$?

A glacier's thickness is $H(t)=80+{\int }_0^{t}g(s)ds$ meters, where $g$ is in m/year. If $g(6)=-0.4$, what does this tell you?