The rate of change of the volume of water in a large industrial storage tank, R(t)R(t)R(t) in m3/day\text{m}^3/\text{day}m3/day, is modeled by the function:
R(t)=54t2+2t−14,t>0 R(t) = \frac{54}{t^2} + 2t - 14, \quad t > 0 R(t)=t254+2t−14,t>0where t t\,t is the time in days since the start of a maintenance cycle. Using calculus:
Find the set of values of t t\,t for which the rate of change R(t)R(t)R(t) is increasing, giving your answer in the form t>ab3t > a\sqrt[3]{b}t>a3b where a a\,a and b b\,b are integers.
Show that ∫39(54t2+2t−14)dt=0\displaystyle \int_{3}^{9} \left( \frac{54}{t^2} + 2t - 14 \right) dt = 0∫39(t254+2t−14)dt=0.
Given that ∫36(54t2+2t−14)dt=−6\displaystyle \int_{3}^{6} \left( \frac{54}{t^2} + 2t - 14 \right) dt = -6∫36(t254+2t−14)dt=−6:
(i) State the value of ∫69(54t2+2t−14)dt\displaystyle \int_{6}^{9} \left( \frac{54}{t^2} + 2t - 14 \right) dt∫69(t254+2t−14)dt.
(ii) Find the value of the constant k k\,k such that ∫36(54t2+2t+k)dt=0\displaystyle \int_{3}^{6} \left( \frac{54}{t^2} + 2t + k \right) dt = 0∫36(t254+2t+k)dt=0.
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.