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3.6 Integration (A-level only)

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Question 118

A geologist models the rate of thermal energy emission, PPP, from a cooling volcanic vent using the equation

P(t)=8+2t3+ln⁡tt4,t≥1 P(t) = \frac{8 + 2t^3 + \ln t}{t^4}, \quad t \ge 1 P(t)=t48+2t3+lnt​,t≥1

where P P\,P is measured in megawatts and t t\,t is the time in hours since initial observation.

a.

Find ∫ln⁡tt4 dt\displaystyle \int \frac{\ln t}{t^4} \,\text{d}t∫t4lnt​dt

[3]
b.

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+ln⁡ba + \ln ba+lnb, where a a\,a and b b\,b are constants to be determined.

[5]
Markscheme

3.6 Integration (A-level only) Questions

  1. A Level
  2. /Maths
  3. /3.6 Integration (A-level only)

363 exam-style questions on CCEA A Level Maths 3.6 Integration (A-level only), covering 3.6.1 Integration (A-level only), 3.6.2 Integration (A-level only), 3.6.3 Integration (A-level only), 3.6.4 Integration (A-level only), 3.6.5 Integration (A-level only), 3.6.6 Integration (A-level only), and 3.6.7 Integration (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.

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