Maths — Pure Mathematics 2 — IAL

14 markseasy

Evaluate

[i] $\int_{1}^{2} 4 x d x ,$

[ii] $\int_{0}^{3} ( 9 x^{2} + 4 x ) d x$ .

Your answer

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24 markseasy

Evaluate

$\int_{- 3}^{4} ( 2 k x + 3 k x^{2} ) d x$

giving your answer in terms of $k$ .

Your answer

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35 markseasy

2 marks

The area bounded by the curve with equation $y=9-x^{2}$ , the $x$ -axis and the vertical lines with equations $x=1$ and $x=2$ is to be found.

Write down an integral that would find this area.

Your answer

3 marks

Evaluate your integral from part [a] and hence find the area described above.

Your answer

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43 markseasy

The diagram below shows the graph of $y=7x-x^{2}-6$

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Find the shaded area, giving your answer as a fraction in its simplest terms.

Your answer

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54 markseasy

The diagram below shows the graph of $y=x^{2}-8x+12$

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Find the shaded area marked $R$ , giving your answer as a fraction in its simplest form.

Your answer

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65 markseasy

1 mark

Simplify $x-(x^{2}-8x+18)$ .

Your answer

4 marks

The diagram below shows the graphs of $y=x^{2}-8x+18$ and $y=x$ .

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Find the shaded area marked $R$ , giving your answer as a fraction in its simplest terms.

Your answer

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72 markseasy

A student is estimating the area bounded by the curve $y=f(x)$ , the $x$ -axis and the lines $x=a\mathrm{and}x=b$ .

The student intends to estimate the area by using trapezia of equal width.

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Add to the diagram above to show how the student can use 4 trapezia to estimate the area.

Your answer

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85 markseasy

3 marks

The graph of $y=8-\sqrt{(3x^{2}+16)}$ is shown below.

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Use the trapezium rule, with four strips (such that $n=4\mathrm{and}h=1)$ , to estimate the shaded area. You may use the values on the graph to help.

Your answer

2 marks

State whether the estimate in part [a] is an under-estimate or an over-estimate, giving a reason for your answer.

Your answer

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98 markseasy

The diagram below shows part of the graph with equation $y = ( x - 2 )^{ \frac{2}{3} }$ .

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The trapezium rule is to be used to estimate the shaded area of the graph which is given by the integral

$\int_{4}^{10} ( x - 2 )^{\frac{2}{3 }} d x.$

[i] All of the values in the table below will be used in the trapezium rule. Write down the number of ordinates that will be used, the number of strips and the width of each strip.

4	5	6	7	8	9	10
1.59	2.08	2.52	2.92	3.30	3.70	4.00

[ii] Apply the trapezium rule, using the values above, to find an estimate of the shaded area.

[iii] State, with a reason, whether your answer to part [ii] is an over-estimate or an under-estimate.

Your answer

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104 marksmedium

Given

$\int_{k}^{5} ( 2 x - 1 ) d x = 20$

find the value of the positive constant $k$ .

Your answer

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112 marksmedium

Evaluate

$\int_{1}^{5} ( 4 x + 6 x^{2} ) d x$

Your answer

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125 marksmedium

1 mark

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

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

Your answer

1 mark

Show that $4-(x-2)^{2}=4x-x^{2}$

Your answer

2 marks

Evaluate

$\int_{0}^{4} ( 4 x - x^{2} ) d x$

Your answer

1 mark

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

Your answer

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133 marksmedium

The diagram below shows part of the graph of $y=x^{2}-4x+3$ .

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

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Your answer

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148 marksmedium

2 marks

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

Your answer

2 marks

Evaluate

$\int_{1}^{3} ( x^{2} - 4 x + 5 ) d x$

Your answer

4 marks

The diagram below shows the graphs of $y=2$ and $y=x^{2}-4x+5$ . Find the exact area of the shaded region $R$ .

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Your answer

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156 marksmedium

2 marks

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

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Work out the $x$ -coordinates of the points labelled $P,Q\mathrm{and}R$ .

Your answer

4 marks

Work out the area of the shaded region.

Your answer

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169 marksmedium

2 marks

The diagram below shows a sketch of the curves with equations

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

6l4stLzX_8-1-sq-7a-medium-edexelmodelling-with-sequences-and-series-

Find the $x$ -coordinates of the intersections of the two graphs.

Your answer

2 marks

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

$\int_{0}^{ \frac{5}{2} } ( 5 x - 2 x^{2} ) d x$

Your answer

5 marks

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

Your answer

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173 marksmedium

Use the two diagrams below to show how rectangles can be used to give an upper and lower bound when estimating the area under a curve using the trapezium rule.

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Your answer

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182 marksmedium

A student is estimating the area bounded by the curve $y=f(x)$ , the $x$ -axis and the lines $x=a$ and $x=b$ .

The student intends to find the area of two rectangles of equal width in order to estimate the area as shown in the diagram below.

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By drawing a sketch, show how the student’s estimate of the area can be improved while still using rectangles of equal width.

Your answer

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1910 marksmedium

4 marks

The diagram below shows the graph with equation $y=\sqrt{x}+4$ .

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The shaded area is to be estimated using the trapezium rule where $h=2$ .

[i]

Write down the number of strips to be used.

[ii]

Write down the number of ordinates to be used.

[iii]

Complete the table of values for $y=\sqrt{x}+4$ , giving values to three significant figures where appropriate.

$x$	4
$y$

Your answer

4 marks

Use the trapezium rule with all the values from the table above to find an estimate of the integral

$\int_{4}^{16} ( \sqrt{x} + 4 ) d x$

giving your answer to three significant figures.

Your answer

2 marks

State, with a reason, whether your answer to part [b] is an overestimate or an underestimate.

Your answer

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208 marksmedium

5 marks

The trapezium rule is to be used to estimate the integral

$\int_{- 2}^{0} x^{3} + 8 d x$

By completing the table of values below, use the trapezium rule to estimate the integral given above.

$x$	-2	-1.5	-1	-0.5	0
$y=x^{3}+8$

Your answer

3 marks

i] By finding the exact value of the integral above, find the percentage error of your estimate from part [a].

ii] Explain how you could increase the accuracy of your trapezium rule estimate.

Your answer

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215 markshard

Use calculus to find the value of

$\int_{4}^{9} \frac{x^{2} + 1}{ \sqrt{x} } d x$

Your answer

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225 markshard

Given

$\int_{1}^{p}$ ( $1+\frac{1}{x^{2}}$ ) $dx=\frac{15}{4}$

find the value of the constant $p$ , where $p>0$ .

Your answer

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234 markshard

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

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

Your answer

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248 markshard

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

Find the total area of the two shaded regions.

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Your answer

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2511 markshard

5 marks

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.

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

Your answer

6 marks

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.

Your answer

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2611 markshard

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

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Find the exact area of the shaded region.

Your answer

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277 markshard

2 marks

The diagram below shows a sketch of the curves with equations

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

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

Your answer

5 marks

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

Your answer

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286 markshard

4 marks

The trapezium rule is to be used to find an estimate for the integral

$\int_{4}^{8} \mathrm{f} ( x ) d x$

The table below shows values for $x$ and $\mathrm{f} ( x )$ , rounded to three significant figures where appropriate.

$x$	4	4.5	5	5.5	6	6.5	7	7.5	8
$\mathrm{f} ( x )$	3.16	3.39	3.61	3.81	4	4.18	4.36	4.53	4.69

Using the values in the table find

[i]

an estimate for the integral using 2 strips,

[ii]

an estimate for the integral using 4 strips,

[iii]

an estimate for the integral using 8 strips.

Your answer

2 marks

Justify which of the estimates from part [a] will be the most accurate estimate for the integral.

Your answer

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295 markshard

The diagram below shows part of the graph with equation $y=3+2x-x^{2}$

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Use the trapezium rule with 5 strips to find an estimate for the shaded area, giving your answer to three significant figures.

Your answer

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306 markshard

4 marks

Use the trapezium rule with $h=0.25$ to find an estimate for the area bounded by the curve with equation $y=1+2^{x}$ , the $x$ -axis and lines with equations $x=1\mathrm{and}x=2$ . Give your answer to five significant figures.

Your answer

2 marks

The integral

$\int_{1}^{2} ( 1 + 2^{x} ) d x$

can be evaluated as 3.8854 to five significant figures. Using this as its exact value, calculate the percentage error of your estimate from part [a].

Your answer

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315 marksvery_hard

Use calculus to find the value of

$\int_{2}^{4} \frac{x^{3} + \sqrt[3]{x} }{2 \sqrt{x} } d x$

giving your answer correct to 3 significant figures.

Your answer

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325 marksvery_hard

Given

$\int_{q}^{4 q} 5 x \sqrt{x} d x = 15 066$

find the value of the constant $q$ .

Your answer

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335 marksvery_hard

4 marks

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

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

Your answer

1 mark

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

Your answer

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348 marksvery_hard

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.

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Your answer

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3511 marksvery_hard

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

Find the exact area of the shaded region.

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Your answer

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3611 marksvery_hard

The diagram below shows the graphs of $y=16-2x$ and $y=x^{3}-15x^{2}+66x-80$ .

Find the total area of the shaded regions.

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Your answer

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378 marksvery_hard

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

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

Your answer

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387 marksvery_hard

3 marks

The diagram below shows a sketch of the graph of $y=x(x-6)^{2}$ $x\geq0$

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The graph has a local maximum point at $(2,32)$ as indicated on the diagram.

Use the trapezium rule with 5 ordinate values to estimate the area shaded.

Your answer

3 marks

Using the appropriate working values from part [a], find an upper and lower bound for the area shaded.

Your answer

1 mark

Suggest a reason why using the trapezium rule in this case is not appropriate.

Your answer

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396 marksvery_hard

4 marks

The diagram below shows the graph of

$y=\frac{4-x^{2}}{\sqrt{x+2}},x>-2.$

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Use the trapezium rule with six strips to find an estimate of the integral

$\int_{- 1}^{1} ( \frac{4 - x^{2}}{ \sqrt{x + 2} } ) d x$

to five significant figures.

Your answer

2 marks

By using your calculator to find the exact value of the integral to five significant figures, find the percentage error of your estimate from part [a].

Your answer

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