Further Integration (Cambridge (CIE) A Level Maths: Pure 1): Exam Questions

The diagram below shows part of the graph of $y=4-{\left(x-2\right)}^2$.

Write down the values of $x$ where $y=0$.

Show that

$4-{\left(x-2\right)}^2=4x-x^2$

Evaluate

${\int }_0^4\left(4x-x^2\right) \text{d}x$

Write down the area of the region labelled $R$.

Find the $x$-coordinates of the points of intersection of the line with equation $y=2$ and the curve with equation $y=x^2-4x+5$.

Evaluate

${\int }_1^3\left(x^2-4x+5\right) \text{d}x$

The diagram below shows the graphs of $y=2$ and $y=x^2-4x+5$.

Find the exact area of the shaded region $R$.

The diagram below shows the graphs of the line $y=6-x$ and the curve $y=x^2$.

Find the $x$-coordinates of the points labelled $P$, $Q$ and $R$.

Find the area of the shaded region.

The diagram below shows a sketch of the curves with equations

$y=x^2-3x+4$ and $y=4-x^2+2x$

Find the $x$-coordinates of the points of intersection of the two graphs.

Show that the area of the shaded region labelled $R$ is given by

${\int }_0^{\frac{5}{2}}\left(5x-2x^2\right) \text{d}x$

Use calculus to find the area of the shaded region labelled $R$.

The diagram below shows the graph of the curve with equation $y=4-x^2$.

(i) Find the $x$-coordinates of the points where the graph of $y=4-x^2$ crosses the $x$-axis.

(ii) The shaded region $R$ is rotated through 360° about the $x$-axis. Find the volume of the solid formed.

The diagram below shows the graphs of two horizontal lines and the function $y=\text{f}\left(x\right)$.

$\text{f}\left(x\right)=a{x}^2+b$, where $a$ and $b$ are constants.

Point $P$ has coordinates $\left(1,5\right)$.

Point $Q$ has coordinates $\left(2,8\right)$.

(i) Find the values of $a$ and $b$.

(ii) Write down the equations of the two horizontal lines.

The shaded region $R$ is rotated through 360° about the $y$-axis. Find the volume of the solid formed.

The diagram below shows part of the graph of $y=2x+3x^2-2x^3$.

Find the area of the shaded region labelled $R$.

The diagram below shows part of the graph of $y=x\left(x-1\right)\left(x+2\right)$.

Find the total area of the two shaded regions.

The line with equation $5y=14-3x$ cuts the curve with equation $5y=20-2x-x^2$ at the points $P$ and $Q$, as shown.

Find the $x$- and $y$-coordinates of the points $P$ and $Q$.

Find the exact area of the region labelled $R$, giving your answer in the form $\frac{a}{b}$, where $a$ and $b$ are integers to be found.

The diagram below shows the graphs of $y=x+2$ and $y=10x-x^2-16$.

Find the exact area of the shaded region.

The diagram below shows a sketch of the curves with equations

$y=x^3+3$ and $y=-x^3+2x+3$

Find the $x$-coordinates of the points of intersection of the two graphs.

Use calculus to find the total shaded area enclosed by the two graphs.

A mathematical model for a bowl is obtained by rotating through 360° about the $y$-axis the part of the curve $y=\frac{1}{a}{x}^2$ which is between $y=8$ and $y=9$, and then adding a flat bottom. $a$ is a constant such that $a>0$.

Find the capacity of the bowl in terms of $a$.

In the case where the boundaries $y=3$ and $y=6$ are used in place of $y=8$ and $y=9$, the capacity of the bowl is increased by $15\pi$ cubic units.

Find the value of $a$.

What assumption has been made in finding the capacity of the bowl?

The diagram below shows a right-angled triangle with vertices at the origin, the point $\left(h,0\right)$ and the point $\left(h,r\right)$, where $r>0$ and $h>0$.

Find an equation of the line on which the hypotenuse of the right-angled triangle lies, giving your answer in the form $y=\text{f}\left(x\right)$.

The triangle is rotated through 360° about the $x$-axis to form a cone.

Hence use calculus to show that the general formula for the volume $V$ of a cone is

$V=\frac{1}{3}\pi r^{2}h$

where $r$ is the base radius of the cone and $h$ is its perpendicular height.

The diagram below shows part of the curve $C$ defined by the equation $y=\frac{1}{a}{x}^2$, where $a$ is a positive constant. The shaded region $R$ is bounded by the curve, the $x$-axis, and the lines $x=1$ and $x=6$.

Given that the volume of the solid formed when the region $R$ is rotated through 360° about the $x$-axis is $\frac{311\pi }{20}$ cubic units, find the value of $a$.

The diagram below shows part of the graph of $y=x^3-2x^2-x+2$.

Find the total area of the two shaded regions.

The diagram below shows the graphs of $y=12x-x^2-27$ and $y=x^2-14x+45$.

Find the area of the shaded region, $R$.

Starting with the equation of a semicircle of radius $r$, $y=\sqrt{r^2-x^2}$ (where $r>0$), use calculus to show that the general formula for the volume $V$ of a sphere of radius $r$ is

$V=\frac{4}{3}\pi r^3$

The diagram below shows part of the graph of $y=4x-x^3$.

Find the area of the shaded region labelled $R$.

Without doing any additional calculation, explain why ${\int }_{-2}^2\left(4x-x^3\right) \text{d}x$ must be equal to zero.

The diagram below shows the graphs of $4y=4x+17$ and $y=3x^2-8x-16$.

Find the exact area of the shaded region.

Sketch the region bounded by the lines

$x=2$, $x=6$, $2y=x+4$ and $y=p$, where $0<p<2$.

The region described in part (a) is rotated through 360° about the $x$-axis.

Find the volume of the solid formed, giving your answer in terms of $p$.

The solid formed in part (b) will have a 'hole' in its centre.

(i) Find the volume of this 'hole', giving your answer in terms of $p$.

(ii) Hence show that there are no values of $p$ in the given interval that make the volume of the solid equal to the volume of the 'hole'.

The diagram below shows a sketch of the curves with equations $y=x^2+5$ and $y=23-x^2$.

The region $R$ is bounded by the $y$-axis and the two curves. This region is rotated through 360° about the $y$-axis. Find the volume of the solid formed.

The regions marked $R$ and $S$ are now to be considered as a single region bounded by the two curves.

A student calculates the volume swept out when this combined region is rotated through a full 360° about the $y$-axis, without taking the symmetry of the two curves into account. Call this value $I$.

How does $I$ compare to the answer you found in part (a)?

Be sure to fully explain your answer.