A model for the power output PPP (in megawatts) of a tidal turbine over an 11-hour cycle is given by
P(t)=(9t−t2)lnt,1≤t≤9 P(t) = (9t - t^2) \ln t, \quad 1 \le t \le 9 P(t)=(9t−t2)lnt,1≤t≤9where t t\,t is the time in hours after the start of the observation period. The total energy generated during this period is given by the area under the power-time graph.
Show that the total energy generated between t=1t = 1t=1 and t=9t = 9t=9 can be expressed in the form
p+qln3 p + q \ln 3 p+qln3where p p\,p and q q\,q are rational numbers to be determined.
Practise AQA A Level Maths 1.11 H: Integration with exam-style questions for A Level Maths. 436 questions covering 1.11.1 Fundamental Theorem of Calculus, 1.11.2 Integrating standard functions, 1.11.3 Definite integrals and areas, 1.11.4 Integration as the limit of a sum (A-level only), 1.11.5 Integration by substitution and by parts (A-level only), 1.11.6 Integration using partial fractions (A-level only), 1.11.7 Differential equations with separable variables (A-level only), and 1.11.8 Interpreting solutions of differential equations (A-level only), matched to the AQA A Level Maths (7357) 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.
