The cross-section of a custom-designed drainage channel is modeled by the parametric equations
x=2u2+2u,y=10u(7−u),a≤u≤b x = 2u^2 + 2u, \quad y = \dfrac{10}{u(7 - u)}, \quad a \le u \le b x=2u2+2u,y=u(7−u)10,a≤u≤bwhere a a\,a and b b\,b are constants. The surface of the water in the channel is represented by the horizontal line y=1y = 1y=1, which connects the two endpoints of the curve.
Determine the values of the constants a a\,a and bbb, where b>ab > ab>a.
The region R R\,R represents the area of the cross-section of the water, bounded by the curve and the line y=1y = 1y=1. Show that the area of R R\,R is given by
M−k∫ab2u+1u(7−u) du M - k \int_{a}^{b} \frac{2u+1}{u(7-u)} \, du M−k∫abu(7−u)2u+1duwhere M M\,M and k k\,k are constants to be determined.
Express 2u+1u(7−u)\dfrac{2u+1}{u(7-u)}u(7−u)2u+1 in partial fractions.
Use algebraic integration to find the exact area of RRR, giving your answer in the form A−Bln(2.5)A - B \ln(2.5)A−Bln(2.5), where A A\,A and B B\,B are constants.
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.
