An artist is designing a stained-glass panel. The boundary of the panel is modeled by the curve CCC with equation y=6x−x2y = 6x - x^2y=6x−x2, for x≥0x \ge 0x≥0. A decorative lead strip is placed along the line lll with equation y=kxy = kxy=kx, where kkk is a constant and 0<k<60 < k < 60<k<6.
The line lll and the curve CCC intersect at the origin OOO and at the point PPP.
Find, in terms of kkk, the coordinates of PPP.
The region R1R_1R1 is bounded by the curve CCC and the line lll. Show that the area of R1R_1R1 is
(6−k)36 \frac{(6 - k)^3}{6} 6(6−k)3The region R2R_2R2 is bounded by the curve CCC, the xxx-axis, and the line lll, such that R1R_1R1 and R2R_2R2 together comprise the total area under the curve for 0≤x≤60 \le x \le 60≤x≤6. Given that the area of R1R_1R1 is equal to the area of R2R_2R2, find the exact value of kkk.
Practise Edexcel A Level Maths Integration with exam-style questions for A Level Maths. 437 questions covering 11.1 Integrating Standard Functions, 11.2 Integrating f(ax + b), 11.3 Using Trigonometric Identities, 11.4 Reverse Chain Rule, 11.5 Integration by Substitution, 11.6 Integration by Parts, 11.7 Partial Fractions, 11.8 Finding Areas, 11.9 The Trapezium Rule, 11.10 Solving Differential Equations, and 11.11 Modelling with Differential Equations, matched to the Edexcel A Level Maths (9MA0) 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.
