The path of a high-precision laser cutter across a titanium plate is defined by the parametric equations
x=t2+8t,y=12t(8−t),a≤t≤b x = t^2 + 8t, \quad y = \frac{12}{t(8-t)}, \quad a \le t \le b x=t2+8t,y=t(8−t)12,a≤t≤bwhere ttt is the time in seconds, and aaa and bbb are constants. The cutter's path intersects the reference line y=1y = 1y=1 at two points representing the start and end of a specific cooling phase.
Find the value of aaa and the value of bbb, where b>ab > ab>a.
The region RRR is bounded by the path and the line y=1y = 1y=1. Show that the area of region RRR is given by
M−k∫abt+4t(8−t) dt M - k \int_{a}^{b} \frac{t+4}{t(8-t)} \, dt M−k∫abt(8−t)t+4dtwhere MMM and kkk are constants to be found.
Express t+4t(8−t)\frac{t+4}{t(8-t)}t(8−t)t+4 in partial fractions.
Hence, use algebraic integration to find the exact area of RRR, giving your answer in the form P−Qln3P - Q \ln 3P−Qln3, where PPP and QQQ are integers.
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.