The torque τ\tauτ, in Newton-meters, applied to a rotating flywheel is modeled by τ(t)=12tsin(2t)\tau(t) = 12t \sin(2t)τ(t)=12tsin(2t), where ttt is the time in seconds. The total angular impulse JJJ delivered between t=π6t = \frac{\pi}{6}t=6π and t=π4t = \frac{\pi}{4}t=4π is defined by the integral:
J=∫π6π412tsin(2t) dt J = \int_{\frac{\pi}{6}}^{\frac{\pi}{4}} 12t \sin(2t) \, dt J=∫6π4π12tsin(2t)dtGiven that J=aπ+bJ = a\pi + bJ=aπ+b, determine the exact value of the constant aaa and the exact value of the constant bbb.
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.
