A gas flow system has a net rate of production R(t)R(t)R(t) given by R(t)=t4t+9R(t) = t\sqrt{4t+9}R(t)=t4t+9 in m3h−1\text{m}^3\text{h}^{-1}m3h−1, where t t\,t is time in hours.
Use integration by substitution to show that the net volume of gas produced during the interval −2.25≤t≤4-2.25 \le t \le 4−2.25≤t≤4, given by
V=∫−2.254t4t+9 dt V = \int_{-2.25}^{4} t\sqrt{4t+9} \, dt V=∫−2.254t4t+9dtis exactly 31.25 m3.
Fully justify your answer.
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.
