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AS Pure Mathematics · Edexcel 8MA0 · free A-level maths worksheet

Integration

Indefinite integrals of x^n, finding f(x) from f'(x) and a point, definite integrals, areas under and between curves and lines.

12 original questions, 68 marks. Print the worksheet, mark it with the answers sheet, and use the solutions to see every step written out by hand.

Worksheet PDFWorked solutions PDFAnswers PDF

The questions

1

Find ∫ (6x2 − 4x + 5) dx, simplifying each term.

3 marks
Show answer
2x3 - 2x2 + 5x + c
2

Find ∫ (3√x + 4x3) dx, giving each term in its simplest form.

4 marks
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2x3/2 - 2x-2 + c
3

A curve with equation y = f(x) passes through the point (2, 9)

Given that f′(x) = 4x3 − 6x + 1, find f(x)

4 marks
Show answer
f(x) = x4 - 3x2 + x + 3
4

Use algebraic integration to find the exact value of 4∫1 (2x − 3√x) dx

Solutions relying entirely on calculator technology are not acceptable.

4 marks
Show answer
9
5

The curve C has equation y = f(x), x > 0, and passes through the point (1, 2)

Given that f′(x) = (x2 + 2)2x2

  1. show that f′(x) = x2 + A + Bx−2 where A and B are constants to be found
  2. find f(x), simplifying each term.
6 marks
Show answer
a) A = 4, B = 4 b) f(x) = 13x3 + 4x - 4x + 53
6

The curve C has equation y = 5 + 4x − x2

The finite region R is bounded by C and the x-axis.

Use algebraic integration to find the area of R

−2−1123456−2246810OxyRC
5 marks
Show answer
36
7

The curve C has equation y = x2 − 3x

  1. Find the total area of the regions bounded by C, the x-axis and the lines x = 0 and x = 5
  2. Find the value of 5∫0 (x2 − 3x) dx and explain why it is not the same as your answer to part (a).
−1123456−4−224681012OxyC
6 marks
Show answer
a) 796 b) 256; the part below the x-axis (0 < x < 3) gives a negative integral, so it is subtracted instead of added
8

The curve C has equation y = x2 − 2x + 3 and the line l has equation y = x + 7

  1. Find the coordinates of the points where l meets C
  2. The finite region R is bounded by C and l. Use algebraic integration to find the exact area of R
−2−11234502468101214xyRlC
7 marks
Show answer
a) (-1, 6) and (4, 11) b) 1256
9

Given that k > 2 and k∫2 (3x2 − 4) dx = 48

  1. show that k3 − 4k − 48 = 0
  2. Hence show that k = 4 is the only solution.
5 marks
Show answer
a) [x3 - 4x] from 2 to k = k3 - 4k = 48 b) (k - 4)(k2 + 4k + 12) = 0, discriminant -32 < 0, so k = 4 only
10

The curve C has equation y = 12x3, x ≥ 0, and the line l has equation y = 6 − x

  1. Show that l meets C at exactly one point, P(2, 4)
  2. The region R is bounded by C, l and the x-axis. Find the area of R

Solutions relying entirely on calculator technology are not acceptable.

−11234567−11234567OxyRlCP
7 marks
Show answer
a) (x - 2)(x2 + 2x + 6) = 0, the quadratic has no real roots, so only P(2, 4) b) 10
11

The entrance to a road tunnel is modelled by the region between the ground and the curve with equation

y = 6 − 0.06x2

where x metres is the horizontal distance from the centre of the entrance and y metres is the height above the ground.

  1. Find the width of the tunnel at ground level.
  2. Use algebraic integration to find the area of the entrance.
  3. The tunnel is 50 m long and has the same cross-section all the way through. Find the volume of the tunnel.
  4. Give one reason why the model may not give an accurate answer for the amount of space available for traffic.
−12−10−8−6−4−224681012−112345678Oxy
8 marks
Show answer
a) 20 m b) 80 m2 c) 4000 m3 d) e.g. the road, kerbs and lighting take up some of the space, or the roof is not exactly this curve
12

The curve C has equation y = x(k − x), where k is a positive constant.

  1. Show that the area of the finite region bounded by C and the x-axis is k36

The line l with equation y = 2x meets C at the origin O and at the point P.

  1. Find the x-coordinate of P in terms of k
  2. Given that the finite region bounded by C and l has area 36, find the value of k
9 marks
Show answer
a) [kx22 - x33] from 0 to k = k36 b) x = k - 2 c) k = 8

Every question here is original and written by Robinson Tuition. It follows the Edexcel AS specification and suits AQA and OCR AS maths too; this is not an exam-board paper.

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