International A LevelMaths: Pure 2EdexcelExam Questions8. Integration8.1 Integration

Integration (Edexcel International A Level Maths: Pure 2)

Exam Questions

4 hours39 questions
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1
Sme Calculator4 marks

Evaluate 

[i]
integral subscript 1 superscript 2 4 x space d x comma
[ii]
integral subscript 0 superscript 3 left parenthesis 9 x squared plus 4 x right parenthesis d x.

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2
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Evaluate

       integral subscript negative 3 end subscript superscript 4 open parentheses 2 k x plus 3 k x squared close parentheses d x

giving your answer in terms of k.

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3a
Sme Calculator2 marks

The area bounded by the curve with equation  y equals 9 minus x squared, the x-axis and the vertical lines with equations x equals 1 and x equals 2 is to be found. 

Write down an integral that would find this area.

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3b
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Evaluate your integral from part [a] and hence find the area described above.

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4
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The diagram below shows the graph of y space equals space 7 x minus x squared minus 6

8-1-sq-4a-medium-edexelmodelling-with-sequences-and-series-

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

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5
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The diagram below shows the graph of  y space equals space x squared minus 8 x plus 12

8-1-sq-5a-easy-edexelmodelling-with-sequences-and-series-

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

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6a
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Simplify x minus left parenthesis x squared minus 8 x plus 18 right parenthesis.

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6b
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The diagram below shows the graphs of  y equals x squared minus 8 x plus 18 and y equals x.

8-1-sq-6b-easy-edexelmodelling-with-sequences-and-series-

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

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7
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A student is estimating the area bounded by the curve y equals straight f left parenthesis x right parenthesis , the x-axis and the lines x equals a space and space x equals b.

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

8-1-sq-7a-easy-edexelmodelling-with-sequences-and-series-

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

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8a
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The graph of y equals 8 minus square root of left parenthesis 3 x squared plus 16 right parenthesis end root is shown below.

8-1-sq-8a-easy-edexelmodelling-with-sequences-and-series-

Use the trapezium rule, with four strips (such that n equals 4 space and space h equals 1 right parenthesis, to estimate the shaded area. You may use the values on the graph to help.

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8b
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State whether the estimate in part [a] is an under-estimate or an over-estimate, giving a reason for your answer.

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9
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The diagram below shows part of the graph with equation y equals open parentheses x minus 2 close parentheses to the power of 2 over 3 end exponent.

8-1-sq-9a-easy-edexelmodelling-with-sequences-and-series-

 

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

            integral subscript 4 superscript 10 space left parenthesis x minus 2 right parenthesis to the power of fraction numerator 2 over denominator 3 space end fraction end exponent 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.

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This topic pack contains questions on the following:

  • Definite Integration

  • Area Under a Curve

  • Area between a curve and a line

  • Area between two curves

  • Trapezium Rule (Numerical Integration)

1
Sme Calculator4 marks

Given

      integral subscript k superscript 5 left parenthesis 2 x minus 1 right parenthesis d x equals 20

find the value of the positive constant k.

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2
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Evaluate

      integral subscript 1 superscript 5 left parenthesis 4 x plus 6 x squared right parenthesis d x

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3a
Sme Calculator1 mark

The diagram below shows part of the graph of y space equals space 4 minus left parenthesis x minus 2 right parenthesis squared.

8-1-sq-3a-medium-edexelmodelling-with-sequences-and-series-

Write down the values of xwhere y equals 0.

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3b
Sme Calculator1 mark

Show that
      4 minus left parenthesis x minus 2 right parenthesis squared equals 4 x minus x squared

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3c
Sme Calculator2 marks

Evaluate

      integral subscript 0 superscript 4 left parenthesis 4 x minus x squared right parenthesis d x

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3d
Sme Calculator1 mark

Write down the area of the region labelled R.

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4
Sme Calculator3 marks

The diagram below shows part of the graph of  y space equals space x squared minus 4 x plus 3.

Find the area of the shaded region labelled R.

8-1-sq-4a-fig-1-medium-edexelmodelling-with-sequences-and-series-

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5a
Sme Calculator2 marks

Find the x-coordinates of the intercepts of the line with equation  y space equals space 2 and the curve with equation  y space equals space x squared minus 4 x plus 5.

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5b
Sme Calculator2 marks

Evaluate

      integral subscript 1 superscript 3 left parenthesis x squared minus 4 x plus 5 right parenthesis d x

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5c
Sme Calculator4 marks

The diagram below shows the graphs of y space equals space 2  and y space equals space x squared minus 4 x plus 5.
Find the exact area of the shaded region R.

8-1-sq-5c-medium-edexelmodelling-with-sequences-and-series-

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6a
Sme Calculator2 marks

The diagram below shows the graphs of the line y space equals space 6 minus x  and  the curve y space equals space x squared .

8-1-sq-6a-medium-edexelmodelling-with-sequences-and-series-

Work out the x-coordinates of the points labelled P comma Q space and space R.

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6b
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Work out the area of the shaded region.

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7a
Sme Calculator2 marks

The diagram below shows a sketch of the curves with equations

         y space equals space x squared minus 3 x plus 4      and   y space equals space 4 minus x squared plus 2 x

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

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

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7b
Sme Calculator2 marks

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

         integral subscript 0 superscript 5 over 2 end superscript space left parenthesis 5 x minus 2 x squared right parenthesis space d x

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7c
Sme Calculator5 marks

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

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8
Sme Calculator3 marks

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.

8-1-sq-8a-fig-1-medium-edexelmodelling-with-sequences-and-series-

8-1-sq-8a-fig-2-medium-edexelmodelling-with-sequences-and-series-

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9
Sme Calculator2 marks

A student is estimating the area bounded by the curve y equals straight f left parenthesis x right parenthesis, the x-axis and the lines x space equals space a and x space equals space 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.

8-1-sq-9-fig-1-easy-edexelmodelling-with-sequences-and-series-

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

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10a
Sme Calculator4 marks

The diagram below shows the graph with equation y equals square root of x plus 4.

8-1-sq-10a-medium-edexelmodelling-with-sequences-and-series-

The shaded area is to be estimated using the trapezium rule where h equals 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 equals square root of x plus 4, giving values to three significant figures where appropriate.

 x

4

 

 

 

 

 

 

 y

 

 

 

 

 

 

 

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10b
Sme Calculator4 marks

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

         integral subscript 4 superscript 16 left parenthesis square root of x plus 4 right parenthesis space d x

giving your answer to three significant figures.

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10c
Sme Calculator2 marks

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

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11a
Sme Calculator5 marks

The trapezium rule is to be used to estimate the integral

            integral subscript negative 2 end subscript superscript 0 space x cubed plus 8 space 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 space equals space x cubed plus 8 

 

 

 

 

 

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11b
Sme Calculator3 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.

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1
Sme Calculator5 marks

Use calculus to find the value of

            integral subscript 4 superscript 9 fraction numerator x squared plus 1 over denominator square root of x end fraction d x

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2
Sme Calculator5 marks

Given

      integral subscript 1 superscript p( 1 plus 1 over x squared) d x equals 15 over 4

find the value of the constant p, where p greater than 0.

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3
Sme Calculator4 marks

The diagram below shows part of the graph of y space equals space 2 x plus 3 x squared minus 2 x cubed.

8-1-sq-3a-hard-edexelmodelling-with-sequences-and-series-

Find the area of the shaded region labelled R.

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4
Sme Calculator8 marks

The diagram below shows part of the graph of y space equals space x open parentheses x minus 1 close parentheses open parentheses x plus 2 close parentheses.

Find the total area of the two shaded regions.

8-1-sq-4a-hard-edexelmodelling-with-sequences-and-series-

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5a
Sme Calculator5 marks

The line with equation 5 y space equals space 14 minus 3 x cuts the curve with equation  5 y space equals space 20 minus 2 x minus x squared  at the points P and Q, as shown.

8-1-sq-5a-hard-edexelmodelling-with-sequences-and-series-

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

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5b
Sme Calculator6 marks

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

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6
Sme Calculator11 marks

The diagram below shows the graphs of y equals x plus 2 and y equals 10 x minus x squared minus 16.

8-1-sq-6a-hard-edexelmodelling-with-sequences-and-series-

Find the exact area of the shaded region.

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7a
Sme Calculator2 marks

The diagram below shows a sketch of the curves with equations

          y equals x cubed plus 3  and  y equals negative x cubed plus 2 x plus 3

8-1-sq-7a-hard-edexelmodelling-with-sequences-and-series-

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

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7b
Sme Calculator5 marks

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

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8a
Sme Calculator4 marks

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

            integral subscript 4 superscript 8 space straight f left parenthesis x right parenthesis space d x

The table below shows values for x and straight f open parentheses x close parentheses, rounded to three significant figures where appropriate. 

 x

4

4.5

5

5.5

6

6.5

7

7.5

8

 straight f open parentheses x close parentheses

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.

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8b
Sme Calculator2 marks

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

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9
Sme Calculator5 marks

The diagram below shows part of the graph with equation  y equals 3 plus 2 x minus x squared

8-1-sq-9a-hard-edexelmodelling-with-sequences-and-series-

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

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10a
Sme Calculator4 marks

Use the trapezium rule with h equals 0.25 to find an estimate for the area bounded by the curve with equation y equals 1 plus 2 to the power of x, the x-axis and lines with equations x equals 1 space and space x equals 2. Give your answer to five significant figures.

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10b
Sme Calculator2 marks

The integral

         integral subscript 1 superscript 2 space left parenthesis 1 plus 2 to the power of x right parenthesis space 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].

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1
Sme Calculator5 marks

Use calculus to find the value of

            integral subscript 2 superscript 4 space fraction numerator x cubed plus cube root of x over denominator 2 square root of x end fraction d x

giving your answer correct to 3 significant figures.

      

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2
Sme Calculator5 marks

Given

      integral subscript q superscript 4 q end superscript space 5 x square root of x space d x equals 15 space 066

find the value of the constant q.

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3a
Sme Calculator4 marks

The diagram below shows part of the graph of  y equals 4 x minus x cubed.

8-1-sq-3a-very-hard-edexelmodelling-with-sequences-and-series-

Find the area of the shaded region labelled R.

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3b
Sme Calculator1 mark

Without doing any additional calculation, explain why  integral subscript negative 2 end subscript superscript 2 left parenthesis 4 x minus x cubed right parenthesis space d x must be equal to zero.

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4
Sme Calculator8 marks

The diagram below shows part of the graph of  y equals x cubed minus 2 x squared minus x plus 2.

Find the total area of the two shaded regions.

8-1-sq-4a-very-hard-edexelmodelling-with-sequences-and-series-

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5
Sme Calculator11 marks

The diagram below shows the graphs of  4 y equals 4 x plus 17  and  y equals 3 x squared minus 8 x minus 16.

Find the exact area of the shaded region.

8-1-sq-5a-very-hard-edexelmodelling-with-sequences-and-series-

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6
Sme Calculator11 marks

The diagram below shows the graphs of  y equals 16 minus 2 x  and  y equals x cubed minus 15 x squared plus 66 x minus 80.

Find the total area of the shaded regions.

8-1-sq-6a-very-hard-edexelmodelling-with-sequences-and-series-

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7
Sme Calculator8 marks

The diagram below shows the graphs of y equals 12 x minus x squared minus 27 and y equals x squared minus 14 x plus 45.

8-1-sq-7a-very-hard-edexelmodelling-with-sequences-and-series-

Find the area of the shaded region, R.

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8a
Sme Calculator3 marks

The diagram below shows a sketch of the graph of y equals x left parenthesis x minus 6 right parenthesis squared   x greater or equal than 0

8-1-sq-8a-very-hard-edexelmodelling-with-sequences-and-series-

The graph has a local maximum point at left parenthesis 2 space comma 32 right parenthesis as indicated on the diagram. 

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

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8b
Sme Calculator3 marks

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

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8c
Sme Calculator1 mark

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

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9a
Sme Calculator4 marks

The diagram below shows the graph of

            y equals fraction numerator 4 minus x squared over denominator square root of x plus 2 end root end fraction comma space x greater than negative 2.

8-1-sq-9a-very-hard-edexelmodelling-with-sequences-and-series-

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

            integral subscript negative 1 end subscript superscript 1 space open parentheses fraction numerator 4 minus x squared over denominator square root of x plus 2 end root end fraction close parentheses space d x

to five significant figures.

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9b
Sme Calculator2 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].

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