Let $R$ be the first-quadrant region bounded by $y=\sqrt{x}$ and $y=\frac{x}{2}$. Which integral gives the area of $R$? Do not evaluate.

The region bounded by $y=3x-x^2$ and the $x$-axis is revolved about the $x$-axis. Which integral gives its volume? Do not evaluate.

The region enclosed by $x=y^2$ and $x=2y$ is measured with respect to $y$. Which integral gives its area? Do not evaluate.

A region under $y=\sqrt{x}$ above the $x$-axis for $0\le x\le 4$ is revolved about the $x$-axis. Which integral gives the volume? Do not evaluate.

The region bounded by $y=x^2$, $y=4$, and the $y$-axis is revolved about the $y$-axis. Which integral gives the volume? Do not evaluate.

The first-quadrant region under $y=4-x^2$ and above the $x$-axis is the base of a solid. Cross sections perpendicular to the $x$-axis are squares. Which integral gives the volume? Do not evaluate.

The region enclosed by $x=y^2$ and $x=4$ is revolved about the $x$-axis. Which integral gives the volume using disks with respect to $x$? Do not evaluate.

The shaded region in the figure lies between $y=x$ and $y=x^2$. What is its area?

The shaded first-quadrant region in the figure is the base of a solid with square cross sections perpendicular to the $x$-axis. Which integral gives the volume?

The shaded region in the figure is the base of a solid. Cross sections perpendicular to the $x$-axis are semicircles. Which integral gives the volume?

The shaded region in the figure is revolved about the $x$-axis. What is the volume, in cubic units?

The shaded region in the figure is revolved about the $y$-axis. Which shell-method integral gives the volume?

The shaded region in the figure is the base of a solid with equilateral-triangle cross sections perpendicular to the $x$-axis. Which integral gives the volume?

The shaded region in the figure is revolved about the $y$-axis. What is the resulting volume, in cubic units?

Water enters a tank at $r(t)=12+2t$ liters per minute for $0\le t\le 4$ minutes. How much water enters during this interval?

A robot's velocity is $v(t)=3t^2-2t$ meters per second for $0\le t\le 3$ seconds. What is its displacement over the interval?

A pump delivers liquid at $r(t)=4+t$ liters per minute for $0\le t\le 5$ minutes. What does ${\int }_0^{5}r(t)dt$ represent?

A sculpture has cross-sectional area $A(x)=4+x$ square meters at position $x$ meters, for $0\le x\le 6$. What is its volume?

The temperature of a material changes at $R(t)=6-\frac{t}{2}$ degrees Celsius per hour for $0\le t\le 4$. By how many degrees Celsius does its temperature change?

A vertical mural has horizontal width $w(y)=2y+1$ meters at height $y$ meters for $0\le y\le 3$. What is the mural's area?

A greenhouse has temperature $T(t)=18+4t-\frac{t^2}{2}$ degrees Celsius for $0\le t\le 4$ hours. What is its average temperature, to the nearest tenth?

A pump's rate is $r(t)=8+2t$ liters per minute for $0\le t\le 4$. What is its average delivery rate?

A cart moves with speed $v(t)=3+t^2$ meters per second for $0\le t\le 3$ seconds. What is its average speed?

Water depth is $d(x)=5-\frac{x^2}{16}$ meters along a channel for $0\le x\le 8$ meters. What is the average depth?

A trail's width is $W(x)=4+\frac{x}{2}$ meters for $0\le x\le 8$ meters. What is the average width?

A student writes: “The area between $y=x^2-1$ and the $x$-axis on $[-1,1]$ is ${\int }_{-1}^1(x^2-1)dx=-\frac{4}{3}$.” Is the work correct?

A student sets up the volume formed by revolving the region under $y=2x$ on $[0,3]$ about the $x$-axis as $\pi {\int }_0^{3}2xdx$. Is the set-up correct?

A student says ${\int }_0^4(8+2t)dt=48$ liters per minute is the average rate of a pump. Is the statement correct?

For the region bounded by $x=y^2$ and $x=2y$, a student writes ${\int }_0^2(y^2-2y)dy$ for its area. Is the set-up correct?

A continuous function $f$ is defined on $[0,4]$. A student writes, “By the Mean Value Theorem for Integrals, there is a number $c$ in $[0,4]$ such that $f(c)=\frac{1}{4}{\int }_0^{4}f(x)dx$.” Is the justification valid?