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1.11 H: Integration

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Question 329

The figure shows

  • the curve CCC with equation y=2x−x2y = 2x - x^2y=2x−x2
  • the line lll with equation y=mxy = mxy=mx, where mmm is a constant and 0<m<20 < m < 20<m<2

The line and the curve intersect at the origin OOO and at the point PPP.

Coordinate diagram showing the parabola C, the line l through O and P, and hatched regions R1 and R2.

a.

Find, in terms of mmm, the coordinates of PPP.

[2]
b.

The region R1R_1R1​, shown shaded in the figure, is bounded by CCC and lll.

Show that the area of R1R_1R1​ is

(2−m)36 \frac{(2 - m)^3}{6} 6(2−m)3​
[3]
c.

The region R2R_2R2​, also shown shaded in the figure, is bounded by CCC, the xxx-axis and lll. Given that the area of R1R_1R1​ is equal to the area of R2R_2R2​,

find the exact value of mmm.

[3]

1.11 H: Integration Questions

  1. A Level
  2. /Maths
  3. /1.11 H: Integration

Practise AQA A Level Maths 1.11 H: Integration with exam-style questions for A Level Maths. 436 questions covering 1.11.1 Fundamental Theorem of Calculus, 1.11.2 Integrating standard functions, 1.11.3 Definite integrals and areas, 1.11.4 Integration as the limit of a sum (A-level only), 1.11.5 Integration by substitution and by parts (A-level only), 1.11.6 Integration using partial fractions (A-level only), 1.11.7 Differential equations with separable variables (A-level only), and 1.11.8 Interpreting solutions of differential equations (A-level only), matched to the AQA A Level Maths (7357) specification and written in Paper 1, Paper 2 and Paper 3 style. Every question includes a full worked solution and mark scheme, so you can see where marks are awarded rather than just whether you got the answer right.

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