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.
438 exam-style questions on OCR A Level Maths 1.8 Integration, covering 1.8.1 Fundamental theorem of calculus (A-level only), 1.8.2 Integrating x^n, 1.8.3 Integrating standard functions (A-level only), 1.8.4 Evaluating definite integrals, 1.8.5 Area between a curve and the x-axis, 1.8.6 Area between two curves, 1.8.7 Integration as the limit of a sum (A-level only), 1.8.8 Integration by substitution (A-level only), 1.8.9 Integration by parts (A-level only), 1.8.10 Use of partial fractions in integration (A-level only), 1.8.11 Differential equations with separable variables (A-level only), 1.8.12 Interpreting the solution of a differential equation (A-level only), and 1.8 Integration. Each one has a worked solution and a mark scheme showing where the marks go.