Applications of Integration (Edexcel IGCSE Further Pure Maths): Exam Questions

Show that $\cos (A–B)–\cos (A+B)=2\sin A\sin B$

Hence express $2\sin 5x\sin 3x$ in the form $\cos mx–\cos nx$where $m$ and $n$ are integers, giving the value of $m$ and the value of $n$,

(i) Find $\int 4\sin 5\theta \sin 3\theta \mathrm{d}\theta$

(ii) Hence evaluate ${\int }_0^{\frac{\mathrm{\pi }}{6}}4\sin 5\theta \sin 3\theta \mathrm{d}\theta$, giving your answer in the form $\frac{a\sqrt{b}}{c}$ where $a$, $b$ and $c$ are integers.

$\mathrm{f}\left(x\right)=7+4x–2x^2$

Given that $\mathrm{f}\left(x\right)$ can be written in the form $P{(x+Q)}^2+R$ where $P,Q$ and $R$ are constants,

find the value of $P$, the value of $Q$and the value of $R$.

(i) Hence write down the maximum value of $\mathrm{f}\left(x\right)$,

(ii) Write down the value of $x$ for which this maximum occurs.

The curve $C$ has equation $y=7+4x–2x^2$

The line $l$ with equation $y=4–x$ intersects $C$ at two points.

Find the $x$ coordinates of these two points.

The finite region bounded by the curve $C$ and the line $l$ is rotated $360°$ about the $x$-axis.

Use algebraic integration to find, to 3 significant figures, the volume of the solid generated.

Figure 3 shows part of the curve $C$ with equation $y=\frac{1}{4x},x>0$ and part of the curve $S$ with equation $y=2x^2,x⩾0$

The curve $C$ and the curve $S$ intersect at the point $A$

Find the coordinates of point $A$

The finite region $R$ , shown shaded in Figure 3, bounded by the curve $C$, the curve $S$ and the straight line $y=4$ is rotated through 360º about the $y$-axis.

Find, using algebraic integration, the exact volume of the solid formed.

Figure 3 shows part of the curve $C$ with equation $y^2=x-1$and part of the line $l$ with equation $2y+x-4=0$

The region $R$, bounded by the $x$-axis, the curve $C$ and the line $l$, is rotated through 360° about the $x$-axis.

Using algebraic integration, find the exact value of the volume of the solid generated.

Using a formula (opens in a new tab), show that

$\cos 2\theta =2{\cos }^2\theta -1$

Using a formula (opens in a new tab), show that

$\cos 2\theta =2{\cos }^2\theta -1$

Hence show that

${\int }_{\frac{\pi }{3}}^{\frac{3\pi }{4}}\left(2{\cos }^2\theta -1\right) d\theta =-\frac{a+\sqrt{b}}{c}$

where $a$, $b$ and $c$ are integers to be found.

Figure 3 shows part of the curve $C_1$ with equation $y=2{\cos }^2\theta -1$and part of the curve $C_2$ with equation $y=-\cos \theta$

Point $B$ is the intersection of $C_1$ and $C_2$ as shown in Figure 3

Point $A$$\left(\frac{3\mathrm{\pi }}{4},0\right)$ is the intersection of $C_1$ with the $\theta$-axis as shown in Figure 3

Point $E$ $\left(\frac{\pi }{2},0\right)$ is the intersection of $C_2$ with the $\theta$-axis as shown in Figure 3

The finite region $R$, shown shaded in Figure 3, is bounded by the $\theta$-axis, $C_1$ and $C_2$

Use calculus to find, in its simplest form, the exact area of $R$

$\mathrm{f}\left(x\right)=10+6x-x^2$

Given that $\mathrm{f}\left(x\right)$ can be written in the form $A(x+B{)}^2+C$ where $A$,$B$ and $C$ are constants,

find the value of $A$, the value of $B$ and the value of $C$

$\mathrm{f}\left(x\right)=10+6x-x^2$

Given that $\mathrm{f}\left(x\right)$ can be written in the form $A(x+B{)}^2+C$ where $A$,$B$ and $C$ are constants,

Hence, or otherwise, find

(i) the value of $x$ for which $\mathrm{f}\left(x\right)$ has its greatest value

(ii) the greatest value of $\mathrm{f}\left(x\right)$

The curve $C$ has equation $y=\mathrm{f}\left(x\right)$

The curve $S$ with equation $y=x^2-x+13$ intersects curve $C$ at two points.

Find the $x$ coordinate of each of these two points.

$\mathrm{f}\left(x\right)=10+6x-x^2$

Given that $\mathrm{f}\left(x\right)$ can be written in the form $A(x+B{)}^2+C$ where $A$,$B$ and $C$ are constants,

The curve $C$ has equation $y=\mathrm{f}\left(x\right)$

The curve $S$ with equation $y=x^2-x+13$ intersects curve $C$ at two points.

Use algebraic integration to find the exact area of the finite region bounded by the curve $C$ and the curve $S$

The curve $S$ with equation $y=\frac{x^2}{4}+2$ where $x⩾0$ and the line $l$ with equation $2y-x-4=0$ where $x⩾0$ intersect at the points $A$and $B$, as shown in Figure 2.

(i) Show that the coordinates of point $A$ are (0, 2)

(ii) Find the coordinates of the point $B$

The finite region bounded by $S$ and $l$, shown shaded in Figure 2, is rotated through $2\pi$ radians about the $y$-axis.

Use algebraic integration to find the volume of the solid generated.
Give your answer in terms of $\pi$

Expand ${\left(1+\frac{x}{3}\right)}^{-3}$ in ascending powers of $x$ up to and including the term in $x^3$

Where appropriate express each coefficient as an exact fraction in its lowest terms.

Write down the range of values of $x$ for which your expression is valid.

Express ${\left(3+x\right)}^{-3}$ in the form $P{\left(1+Qx\right)}^{-3}$ where $P$ and $Q$ are rational numbers whose values should be stated.

$\mathrm{f}\left(x\right)=\frac{1+4x}{(3+x{)}^3}$

Obtain a series expansion for $\mathrm{f}\left(x\right)$ in ascending powers of $x$ up to and including the term in $x^2$

$\mathrm{f}\left(x\right)=\frac{1+4x}{(3+x{)}^3}$

Hence, using algebraic integration, obtain an estimate of ${\int }_0^{0.2}\mathrm{f}\left(x\right) \mathrm{d}x$

Give your answer to 5 significant figures.

Figure 2 shows the graph of part of the curve $C$ with equation $y=\sqrt{2x+6}$
The finite region enclosed by the curve $C$ and the straight line with equation $3y-x=3$ is rotated through 360° about the $x$-axis.

Use algebraic integration to find the exact volume of the solid generated.
Give your answer in terms of $\pi$

Expand $(1-8x^2{)}^{-\frac{1}{2}}$ in ascending powers of $x$, up to and including the term in $x^6$ giving each coefficient as an integer.

$\mathrm{g}\left(x\right)=\frac{a+bx}{\sqrt{1-8x^2}}$ where $a$ and $b$ are prime numbers

Given that the fourth and fifth terms, in ascending powers of $x$, in the series expansion of $\mathrm{g}\left(x\right)$ are 20$x^3$and 48$x^4$ respectively,

find the value of $a$ and the value of $b$

Using the first five terms, in ascending powers of $x$, in the series expansion of $\mathrm{g}\left(x\right)$

obtain an estimate, to 4 significant figures, of ${\int }_0^{0.2}\mathrm{g}\left(x\right) \mathrm{d}x$

The region $R$, shown shaded in Figure 2, is bounded by the curve with equation $y=x^2+1$and the curve with equation $x^2+y^2=11$

The two curves intersect at the point $A$ and at the point $B$.

Find the $x$coordinate of the point $A$and the $x$ coordinate of the point $B$.

The region $R$ is rotated through 360° about the $x$‑axis.

Use algebraic integration to find the volume, to 2 decimal places, of the solid generated.

Figure 3 shows part of the curve $C$ with equation $y=4-{\mathrm{e}}^{2x}$
The curve $C$ crosses the $y$-axis at the point $A$and the $x$-axis at the point $B$.

(i) Write down the $y$ coordinate of point $A$.

(ii) Show that the $x$ coordinate of $B$ is $x=\ln 2$

The line $l$ is the normal to $C$ at the point $B$.

Find an equation for $l$ , giving your answer in the form $y=mx+c$

The finite region $R$ is bounded by $C$, $l$ and the $y$-axis.

Using calculus, find the area of $R$.
Give your answer to one decimal place.