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1.8 Integration

1.8 Integration

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Question 305

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]
Markscheme

1.8 Integration Questions

  1. A Level
  2. /Maths
  3. /1.8 Integration

438 exam-style questions on OCR A Level Maths 1.8 Integration, covering 1.8.1 Fundamental theorem of calculus (A-level only), 1.8.2 Integrating x^n, 1.8.3 Integrating standard functions (A-level only), 1.8.4 Evaluating definite integrals, 1.8.5 Area between a curve and the x-axis, 1.8.6 Area between two curves, 1.8.7 Integration as the limit of a sum (A-level only), 1.8.8 Integration by substitution (A-level only), 1.8.9 Integration by parts (A-level only), 1.8.10 Use of partial fractions in integration (A-level only), 1.8.11 Differential equations with separable variables (A-level only), 1.8.12 Interpreting the solution of a differential equation (A-level only), and 1.8 Integration. Each one has a worked solution and a mark scheme showing where the marks go.

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