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.
864 exam-style questions on Edexcel A Level Maths Integration, 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. Each one has a worked solution and a mark scheme showing where the marks go.