A geologist models the rate of thermal energy emission, PPP, from a cooling volcanic vent using the equation
P(t)=8+2t3+lntt4,t≥1 P(t) = \frac{8 + 2t^3 + \ln t}{t^4}, \quad t \ge 1 P(t)=t48+2t3+lnt,t≥1where P P\,P is measured in megawatts and t t\,t is the time in hours since initial observation.
Find ∫lntt4 dt\displaystyle \int \frac{\ln t}{t^4} \,\text{d}t∫t4lntdt
The total energy EEE, in megawatt-hours, emitted by the vent between t=1t = 1t=1 and t=2t = 2t=2 is given by the area under the curve P(t)P(t)P(t) between these two times.
Use your result from part (a) to calculate the exact value of EEE. Give your answer in the form a+lnba + \ln ba+lnb, where a a\,a and b b\,b are constants 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.
