Question

In an AC circuit, the current has the form i (t) = I cos(ωt) for constants I and ω. The power is defined as Ri^2 for a constant
R. Find the average value of the power by integrating over the
interval [0, 2π/ω].

Answer 1

\[i(t)=I\cos{\omega t}\]\[P=\frac{\omega}{2\pi}\int\limits_{0}^{2\pi/\omega}p(t)dt=\frac{\omega R}{2\pi}\int\limits_{0}^{2\pi/\omega}i^2(t)dt=\frac{\omega RI^2}{2\pi}\int\limits_{0}^{2\pi/\omega}\cos^2{\omega t}dt=\]\[=\frac{\omega RI^2}{2\pi}\int\limits_{0}^{2\pi/\omega}\frac{1+\cos{2\omega t}}{2}dt=\frac{\omega RI^2}{2\pi}\frac{\pi}{\omega}=\frac{RI^2}{2}\]