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.
