CIE A Level Maths: Pure 1

Topic Questions

6.2 Further Integration

1a
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2 marks

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

Write down an integral that would find this area.

1b
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3 marks

Evaluate your integral from part (a) and hence find the area described above.

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2
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3 marks

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

screenshot-2023-08-08-at-12-28-34-pm

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

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3
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4 marks

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

screenshot-2023-08-08-at-12-31-42-pm

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

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4a
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1 mark

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

4b
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4 marks

The diagram below shows the graphs of y equals x squared minus 8 x plus 18 space a n d space y equals x. 

screenshot-2023-08-08-at-12-36-32-pm

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

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5
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5 marks

The diagram below shows the region , bounded by the straight lines with equations y equals 2 x plus 1 comma space x equals 2 comma x equals 4 and the -axis.

screenshot-2023-08-08-at-12-39-55-pm

(i)
For y space equals space 2 x space plus space 1 comma space show that y squared space equals space 4 x squared space plus space 4 x space plus space 1.
(ii)
Find the volume of the solid formed when the region R is rotated 360 degree around the -axis.

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6
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5 marks

The diagram below shows the region R, bounded by the curve with equation y equals x squared plus 2 comma the lines y equals 6 space and space y equals 11,and the y-axis.

screenshot-2023-08-08-at-12-48-04-pm

Find the volume of the solid formed when the region R is rotated 360 degree around the y-axis.

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1
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4 marks

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

screenshot-2023-08-09-at-8-38-02-am

Find the area of the shaded region labelled R.

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2
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8 marks

The diagram below shows part of the graph of y equals x left parenthesis x minus 1 right parenthesis left parenthesis x plus 2 right parenthesis.

Find the total area of the two shaded regions.

screenshot-2023-08-09-at-8-40-40-am

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3
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3 marks

The diagram below shows a sketch of the curve with equation y equals 1 plus 2 x minus space 1 fourth x squared.

screenshot-2023-08-09-at-8-44-07-am

The point P open parentheses x space comma space y close parentheses  lies on the curve.

The shaded rectangle shown has width delta x and height y

Calculate

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4a
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5 marks

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

screenshot-2023-08-09-at-9-15-35-am

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

4b
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6 marks

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

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5
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11 marks

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

    screenshot-2023-08-09-at-9-26-07-am

Find the exact area of the shaded region.

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6a
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2 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

screenshot-2023-08-09-at-9-31-40-am

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

6b
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5 marks

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

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7a
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4 marks

A mathematical model for a bowl is obtained by rotating through 360 degree about the y-axis the part of the curve y space equals space 1 over a x squared  which is between y space equals space 8 and  y space equals space 9 and then adding a flat bottom.  a is a constant such that a space greater than space 0.

Find the capacity of the bowl in terms of a.

7b
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3 marks

In the case where the boundaries  y space equals space 3 and y space equals space 6 are used in place of y space equals space 8 and y space space equals space 9,  the capacity of the bowl is increased by 15 straight pi cubic units.
Find the value of a.

7c
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1 mark

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

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8a
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2 marks

The diagram below shows a right-angled triangle with vertices at the origin, the point open parentheses h space comma space 0 close parentheses and the point open parentheses h space comma space r close parentheses,  where r space greater than space 0 and h space greater than space 0.

screenshot-2023-08-09-at-9-49-12-am

Find an equation of the line on which the hypotenuse of the right-angled triangle lies, giving your answer in the form y space equals straight f space open parentheses x close parentheses.

8b
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3 marks

The triangle is rotated 360 degree about the x-axis to form a cone.
Thus use calculus to prove that the general formula for the volume V, of a cone is

      V space equals space 1 third πr squared space straight h.

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

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9
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4 marks

The diagram below shows part of the curve C defined by the equation y space equals space 1 over a x squared comma where a is a positive constant.  The shaded region R is bounded by the curve, the  x-axis, and the lines x space equals space 1 and x space equals space 6. 

screenshot-2023-08-09-at-10-19-59-am

Given that the volume of the solid formed when the region R  is rotated 360 degree about the x-axis is fraction numerator 311 straight pi over denominator 20 end fraction  cubic units, find the value of a.

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This topic pack contains questions on the following:

  • Area Under a Curve

  • Area between a curve and a Line

  • Area between two curves

  • Volumes of Revolution around the x-axis

  • Volumes of Revolution around the y-axis

  • Adding and Subtracting Volumes

  • Modelling with Volumes of Revolution

1a
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1 mark

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

screenshot-2023-08-08-at-9-09-05-pm

Write down the values of x where y space equals space 0.

1b
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1 mark

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

1c
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2 marks

Evaluate

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

1d
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1 mark

Write down the area of the region labelled R.

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2
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3 marks

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

Find the area of the shaded region labelled R.

screenshot-2023-08-08-at-9-18-02-pm

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3a
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2 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.

3b
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2 marks

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

3c
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4 marks

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

screenshot-2023-08-08-at-9-24-54-pm

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4
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2 marks

A student is estimating the area bounded by the curve y space equals space straight f open parentheses straight x close parentheses comma the  x-axis and the lines x space equals space a and x space 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.

screenshot-2023-08-08-at-9-29-27-pm

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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5a
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2 marks

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

screenshot-2023-08-08-at-9-33-24-pm

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

5b
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4 marks

Work out the area of the shaded region.

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6a
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2 marks

The diagram below shows a sketch of the curves with equations

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

screenshot-2023-08-08-at-9-40-35-pm

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

6b
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2 marks

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

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

6c
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5 marks

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

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7
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5 marks

The diagram below shows the graph of the curve with equation y equals 4 minus x squared.

q7-6-2-medium-cie-a-level-maths

(i)
Find the x-coordinates of the points where  the graph of y space equals space 4 space minus space x squaredintercepts the x-axis.
(ii)
The shaded region, R, is to be rotated 360 degree around the  x-axis.
Find the volume of the shape generated.

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8a
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3 marks

The diagram below shows the graphs of two horizontal lines and the function  y space equals space straight f open parentheses x close parentheses.

straight f open parentheses x close parentheses space equals space a x squared space plus space b comma where a space and space b are constants.

screenshot-2023-08-08-at-9-57-36-pm

Point P has coordinates open parentheses 1 comma 5 close parentheses.

Point Q has coordinates open parentheses 2 comma space 8 close parentheses.

(i)
Find the values of a space and space b.
(ii)
Write down the equations of the two horizontal lines. 
8b
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4 marks

The shaded region, R , is to be rotated 360 degree around the -axis.
Find the volume of the shape generated.

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1a
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4 marks

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

screenshot-2023-08-09-at-11-53-16-am

Find the area of the shaded region labelled R.

1b
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1 mark

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

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2
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8 marks

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

Find the total area of the two shaded regions.

screenshot-2023-08-09-at-11-58-52-am

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3
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11 marks

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

Find the exact area of the shaded region.

screenshot-2023-08-09-at-12-01-41-pm

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4
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11 marks

The diagram below shows the graphs of  y space equals space 16 space minus space 2 x and y space equals space x to the power of 3 space end exponent minus space 15 x squared space plus space 66 x space minus space 80.

Find the total area of the shaded regions.

screenshot-2023-08-09-at-12-06-31-pm

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5
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8 marks

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

screenshot-2023-08-09-at-12-10-41-pm

Find the area of the shaded region, R.

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6a
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3 marks

Sketch the region described by the following inequalities

       space x greater or equal than 2 space space space space      x space less or equal than space 6      2 y space less or equal than space x space plus space 4      y space greater or equal than space p comma space w h e r e space 0 less than space p space less than space 2

6b
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4 marks

The region described in part (a) is rotated through 360 degree about the x-axis.
Find the volume of the solid formed, giving your answer in terms of p.

6c
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4 marks

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’.

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7
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5 marks

Starting with the equation of a semicircle of radius r, y space equals space square root of r squared space minus space x squared end root  (where r space greater than space 0 ),  use calculus to prove that the general formula for the volume,V, of a sphere of radius r is

      V space equals space 4 over 3 πr cubed.

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8a
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6 marks

The diagram below shows a sketch of the curves with equations y equals x squared plus 5 space and space y equals 23 minus x squared.

screenshot-2023-08-09-at-12-28-06-pm

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

8b
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2 marks

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

A student has correctly constructed an integral to calculate the volume formed by rotating that combined region through 360 degree about the y-axis, and has correctly calculated its value,I.

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

Be sure to fully explain your answer.

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