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

$\mathrm{f} ( x ) = 2 x^{3} - 13 x^{2} + 12 x + 27$

Figure 3 shows a sketch of the curve with equation $y = \mathrm{f} ( x )$, which passes through the points with coordinates \left(-1 , 0\right), \left(b , 0\right) and \left(a , 0\right) where $0 < b < a$.

Given that $b = 3$ and $a = \frac{9}{2}$, use algebraic integration to determine the exact value of the total area of the shaded regions shown in Figure 3.

Show that $\mathrm{cos}(A--B)--\mathrm{cos}(A+B)=2\mathrm{sin}A\mathrm{sin}B$

Hence express $2\mathrm{sin}5x\mathrm{sin}3x$ in the form $\mathrm{cos}mx--\mathrm{cos}nx$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 integral _ 0 ^ pi / 6 end ^ 4 sin 5 theta sin 3 theta d theta, giving your answer in the form $\frac{a\sqrt{b}}{c}$ where $a$, $b$ and $c$ are integers.

$f(x)=7+4x--2x^{2}$

Given that $f(x)$ 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 $f(x)$,

(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 $360^{\circ}$ 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 $y = 2 x ^{2} , x \text{greater} - \text{than} or \text{slanted} \text{equal} to 0$

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, show that

$\mathrm{cos}2θ=2\mathrm{cos}^{2}θ-1$

Using a formula, show that

$\mathrm{cos}2θ=2\mathrm{cos}^{2}θ-1$

Hence show that

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

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

Figure 3 shows part of the curve $C_{1}$ with equation $y=2\mathrm{cos}^{2}θ-1$and part of the curve $C_{2}$ with equation $y=-\mathrm{cos}θ$

Point $B$ is the intersection of $C_{1}$ and $C_{2}$ as shown in Figure 3

Point $A\quad ( \frac{3 \mathrm{\pi } }{4} , 0 )$ is the intersection of $C_{1}$ with the $θ$-axis as shown in Figure 3

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

The finite region $R$, shown shaded in Figure 3, is bounded by the $θ$-axis, $C_{1}$ and $C_{2}$

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

$f(x)=10+6x-x^{2}$

Given that $\mathrm{f} ( x )$ 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 $C$

$f(x)=10+6x-x^{2}$

Given that $\mathrm{f} ( x )$ 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 $\mathrm{f} ( x )$ has its greatest value

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

The curve $C$ has equation $y=f(x)$

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.

$f(x)=10+6x-x^{2}$

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

The curve $C$ has equation $y=f(x)$

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 greater-than or slanted equal to 0 and the line $l$ with equation $2y-x-4=0$ where x greater-than or slanted equal to 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 $2π$ radians about the $y$-axis.

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

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

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

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

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

Hence, using algebraic integration, obtain an estimate of $\int_{0}^{0.2} \mathrm{f} ( x )   \mathrm{d} x$

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 $π$

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.

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

Given that the fourth and fifth terms, in ascending powers of $x$, in the series expansion of $\mathrm{g} ( x )$ 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 $\mathrm{g} ( x )$

obtain an estimate, to 4 significant figures, of $\int_{0}^{0.2} \mathrm{g} ( x )   \mathrm{d} x$

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 $y=4-e^{2x}$ 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 $x=\mathrm{ln}2$

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.