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.
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.