A mechanical system's variable torque τ\tauτ, in Newton-metres, is modeled as a function of its angular displacement θ\thetaθ (in radians) by the equation τ(θ)=14θsin(12θ)\displaystyle \tau(\theta) = \frac{1}{4}\theta \sin\left(\frac{1}{2}\theta\right)τ(θ)=41θsin(21θ). Prove that the total work done W=∫02πτ(θ) dθW = \int_{0}^{2\pi} \tau(\theta) \, d\thetaW=∫02πτ(θ)dθ is equal to exactly π \pi\,π Joules.
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.
