Find the area of the shaded region on the diagram.

Find the value of $b$.

This region is now rotated through $2\pi$ radians about the $x$-axis. Find the volume of revolution.

Write down the coordinates of the vertex of the graph of g.

Find the volume of the solid formed. Give your answer in the form $\frac{\mathrm{\pi }}{4}\left(a+b \ln \left(c\right)\right)$, where $a, b, c\in \mathrm{\mathbb{Z}}$.

Find $\int {\left(f\left(x\right)\right)}^2\text{d}x$.

The shape of a vase is formed by rotating a curve about the $y$-axis.

The vase is $10 \text{cm}$ high. The internal radius of the vase is measured at $2 \text{cm}$ intervals along the height:

Use the trapezoidal rule to estimate the volume of water that the vase can hold.

Find the area enclosed by $y=f\left(x\right)$, the $x$-axis and the line $x=0.5$.

Write down the maximum and minimum value of $v$.

Write down the coordinates of A.

Calculate the area of $A$.

The function $f$ is defined by $f\left(x\right)={\left(x-1\right)}^2$, $x$ ≥ 1 and the function $g$ is defined by $g\left(x\right)=x^2+1$, $x$ ≥ 0.

The region $R$ is bounded by the curves $y=f\left(x\right)$, $y=g\left(x\right)$ and the lines $y=0$, $x=0$ and $y=9$ as shown on the following diagram.

The shape of a clay vase can be modelled by rotating the region $R$ through 360˚ about the $y$-axis.

Find the volume of clay used to make the vase.

On the grid above, sketch the graph of g for −2 ≤ x ≤ 4.

Write down the rate of change of $f$ at A.

Write down the value of $a$.

Factorize $x^2+3x+2$.

Find the coordinates of B.

Sketch the graph of $y=f\left(\left|x\right|\right)$.

Given that $p\left(t\right)$ can be written as $p\left(t\right)=a\text{sin}\left(b\left(t-c\right)\right)+d$ where $a$, $b$, $c$, $d$ > 0, use your graph to find the values of $a$, $b$, $c$ and $d$.

Sketch the graph of $y=p\left(t\right)$ for 0 ≤ $t$ ≤ 0.02 , showing clearly the coordinates of the first maximum and the first minimum.

The empty barrel is being filled with water. The volume $V{\text{m}}^3$ of water in the barrel after $t$ minutes is given by $V=0.8(1-{\text{e}}^{-0.1t})$. How long will it take for the barrel to be half-full?

The region enclosed by the graph of $f$, the y-axis and the x-axis is rotated 360° about the x-axis.

Find the volume of the solid formed.

The region enclosed by the graph of $f$, the $x$-axis and the $y$-axis is rotated 360º about the $x$-axis. Find the volume of the solid formed.

Hence find the value of $p$ if ${\int }_0^{1}f(x)\text{d}x=\ln (p)$.

The region R, is bounded by the graph of the function found in part (b), the x-axis, and the lines $x=1$ and $x=\alpha$ where $\alpha >1$. The area of R is $\sqrt{2}$.

Find the value of $\alpha$.

Find the area of the region enclosed by the graphs of f and g.

Find the value of $p$.

Show that the $x$-coordinate of the minimum point on the curve $y=f(x)$ satisfies the equation $\tan x=2x$.

Find the $x$-intercept of the graph of $f$.

Express $x^2+3x+2$ in the form $(x+h{)}^2+k$.

When $t$ = 0, the volume of water in the container is 2.3 m³. It is known that the container is never completely full of water during the 4 hour period.

Find the minimum volume of empty space in the container during the 4 hour period.

Write down a definite integral to represent the area of $A$.

Find $p_{av}$(0.007).

Find the coordinates of the point on the graph of $f$ where the normal to the graph is parallel to the line $y=-x$.

Sketch the graph of $y=f(x)$ showing clearly the minimum point and any asymptotic behaviour.

Determine the values of $x$ for which $f(x)$ is a decreasing function.

Show that $\frac{1}{x+1}-\frac{1}{x+2}=\frac{1}{x^2+3x+2}$.

Show that there are exactly two points on the curve where the gradient is zero.

Show that $\text{lo}{\text{g}}_{r^2}x=\frac{1}{2}\text{lo}{\text{g}}_{r}x$ where $r,x\in {\mathbb{R}}^{+}$.

Let $R$ be the region enclosed by the graph of $f$ , the $x$-axis, the line $x=b$ and the line $x=a$. The region $R$ is rotated 360° about the $x$-axis. Find the volume of the solid formed.

Find the value of $p$.

Find the volume of the container.

Part of the graph of f is shown in the following diagram.

The shaded region R is enclosed by the graph of f, the x-axis, and the lines x = 1 and x = 9 . Find the volume of the solid formed when R is revolved 360° about the x-axis.

Find the x-intercept of the graph of $f$.

Find $\int \left(6x^2-3x\right)\text{d}x$.

The shaded region is rotated by 2$\pi$ about the $y$-axis. Find the volume of the solid formed.

During the interval $p$ < $t$ < $q$, he volume of water in the container increases by $k$ m³. Find the value of $k$.

The following diagram shows part of the graph of $f$.

The region enclosed by the graph of $f$, the $x$-axis and the lines $x=-p$ and $x=p$ is rotated 360° about the $x$-axis. Find the volume of the solid formed.

Express $y$ in terms of $x$. Give your answer in the form $y=p{x}^q$, where p , q are constants.

Write down two transformations that will transform the graph of $y=v\left(t\right)$ onto the graph of $y=i\left(t\right)$.

Find the total time in the interval 0 ≤ $t$ ≤ 0.02 for which $p\left(t\right)$ ≥ 3.

Find the value of $a$.

Find the area of the region enclosed by the graph of $f$, the x-axis and the lines x = 1 and x = 2 .

Find the value of Charlotte’s estimate.

Use the model to find the volume of the barrel.

The outer dome of a large cathedral has the shape of a hemisphere of diameter 32 m, supported by vertical walls of height 17 m. It is also supported by an inner dome which can be modelled by rotating the curve $y=33-0.08x^3$ through 360° about the $y$-axis between $y$ = 0 and $y$ = 33, as indicated in the diagram.

Find the volume of the space between the two domes.

Find the value of $p$ and of $q$.

The normal at P cuts the curve again at the point Q. Find the $x$-coordinate of Q.

Find the equation of the normal to the curve at the point P.

With reference to your graph of $y=p\left(t\right)$ explain why $p_{av}\left(T\right)$ > 0 for all $T$ > 0.

Determine the area of the region enclosed between the graph of $y=f\left(\left|x\right|\right)$, the $x$-axis and the lines with equations $x=-1$ and $x=1$.

Sketch the graph of $f(x)$, indicating on it the equations of the asymptotes, the coordinates of the $y$-intercept and the local maximum.

Find the the rate of change of $f$ at B.

Write down an expression for her estimate of the volume as a sum of two integrals.

Show that the volume of water, $V$, in terms of $h$ is $V=3\mathrm{\pi }\left({\mathrm{e}}^{\frac{h}{3}}-1\right)$.

Hence find the maximum capacity of the bowl in ${\text{cm}}^3$.