The region bounded by the curve $y=\sqrt{4-x}$, the $x$-axis and the $y$-axis is rotated through $2\pi$ radians about the $x$-axis.

A surveyor measures the width of a lake at $25$ metre intervals along a straight baseline of length $150$ metres. The measurements are shown in the table.

The mass of a substance in a container increases at a rate

grams per hour, where $0\leq t\leq1$. Initially, the container holds $10$ grams of the substance.

For $x\geq0$, a rate is modelled by

The curve $y=(x-1)(x-4)$ is considered on the interval $0\leq x\leq5$.

A region is bounded by the curve

the $y$-axis and the horizontal lines $y=0$ and $y=\ln4$. The region is rotated through $2\pi$ radians about the $y$-axis.

The curve $x=y^2-4$ and the $y$-axis enclose a finite region.

The height of a decorative canopy above a horizontal walkway is modelled by

where $0\leq x\leq 8$, and $x$ and $h$ are measured in metres.

During a fundraising event, the total amount collected, in hundreds of dollars, is $Q(t)$, where $t$ is the number of hours after the event begins. The rate of collection is modelled by

for $0\leq t\leq6$. At the start of the event, $Q=25$.

The vertical cross-section of a theatre backdrop is bounded by the $x$-axis, the lines $x=0$ and $x=20$, and the curve

$y=-0.04x^2+0.8x+1$

where both coordinates are measured in metres.

The electrical power produced by a small wind turbine is recorded every $0.5$ hours during a three-hour test. The results are shown in the table.

A survey team measures the width of a coastal wetland at $40$ metre intervals along a baseline of length $240$ metres. The measured widths are shown in the table.

A manufacturer cuts a panel whose height is modelled by

for $0\leq x\leq20$, where both $x$ and $p(x)$ are measured in metres. The panel lies between the curve and the $x$-axis. For part (b), a vertical cut at $x=a$ divides the panel into two regions of equal area.

The number of people in an exhibition hall is $N(t)$, where $t$ is measured in minutes after the doors open. The net rate of change is modelled by

for $0\leq t\leq10$. Initially, there are $120$ people in the hall.

A curve has gradient

$\frac{dy}{dx}=6x-2$

and passes through the point $(0,2)$. The region between the curve, the $x$-axis and the lines $x=0$ and $x=2$ is denoted by $R$.

A treatment system removes a contaminant from a vessel at the rate

$r(t)=40t e^{-t^2/4}$

grams per hour, where $t\geq0$. Initially, the vessel contains $120$ grams of the contaminant.

The profile of a rotationally symmetric wooden ornament is modelled by

$y=2+\sqrt{x},\qquad 0\leq x\leq9$

where $x$ and $y$ are measured in centimetres. The region under the curve is rotated through $2\pi$ radians about the $x$-axis.

A function $F$ is defined for $x>0$ and satisfies

$F'(x)=x^{-1/2}+\frac{3}{x}$

It is given that $F(1)=2$.

A filtration system removes a contaminant at a rate
$r(t)=12e^{-0.3t}+3$
where $r$ is measured in milligrams per minute and $t$ is the time, in minutes, after the system starts. The term $3$ represents a constant background removal rate.

The rate of change of the volume of water in a tidal basin is modelled by
$q(t)=6\sin\left(\frac{\pi t}{6}\right),\qquad 0\leq t\leq12$
where $q$ is measured in thousands of cubic metres per hour. Positive values represent inflow and negative values represent outflow.

The inside profile of a rotationally symmetric vessel is modelled by
$x=0.8+0.12y^2,\qquad 0\leq y\leq5$
where $x$ and $y$ are measured in metres. The region between the profile and the $y$-axis is rotated through $2\pi$ radians about the $y$-axis.

The accumulated quantity of a dissolved mineral is modelled from the rate
$r(t)=\frac{18t}{t^2+4},\qquad t\geq0$
where $r$ is measured in grams per hour.

The electrical power generated by a solar installation is modelled by
$P(t)=5+3\cos\left(\frac{\pi(t-2)}6\right),\qquad0\leq t\leq12$
where $P$ is measured in kilowatts and $t$ is measured in hours.

The curve

$y=2\sin x-1$

is considered on the interval $0\leq x\leq2\pi$.

A designer models the outer profile of a lampshade by

$x=1+2e^{-y/4},\qquad 0\leq y\leq8$

where $x$ and $y$ are measured in centimetres. The region between the curve and the $y$-axis is rotated through $2\pi$ radians about the $y$-axis.

The curve

$y=x^3-4x$

is considered on the interval $-3\leq x\leq3$.

For $0\leq x\leq\pi$, define

$f(x)=\frac{4\sin x}{3-\cos x}$

The accumulated value from $0$ to $b$ is denoted by

$A(b)=\int_0^b f(x)\,dx$

A decorative solid is formed by rotating the region under
$y=\sqrt{x}e^{-x/2},\qquad 0\leq x\leq a$
about the $x$-axis, where $a>0$.