Answer to Question #187524 in Calculus for Idris

Question

For the function v = 40sin calculate the mean and rms over a range of

Answer 1

Let function is $v = 40sin\omega t$

Mean value over interval 0 to T where $\omega = \frac{2 \pi}{T}$

$V_{mean} = \frac{\int_0^T vdt}{\int_0^T dt} = \frac{\int_0^T 40sin\omega t dt}{\int_0^T dt} = \frac{\frac{40}{\omega}[-cos\omega t]_0^T}{t_0^T}$

$V_{mean} = \frac{ -40[cos\omega T - cos0]}{\omega T} = -\frac{40}{\omega T}[cos2\pi - cos0] = -\frac{40}{\omega T}[1-1] = 0$

RMS value in interval 0 to T,

$V_{rms} = \sqrt{\frac{\int_0^T v^2 dt}{\int_0^T dt}}$

$V_{rms} = \sqrt{\frac{\int_0^T 1600sin^2\omega t dt}{\int_0^T dt}} = \sqrt{\frac{\int_0^T 1600 \frac{(1-cos2\omega t)}{2}dt}{\int_0^T dt} }$

$V_{rms} = \sqrt{\frac{ 800(t - \frac{1}{2\omega}sin2\omega t)_0^T }{t_0^T}} = \sqrt{\frac{ 800(T - \frac{1}{2\omega}(sin2\omega T - sin0)) }{T} }$

$V_{rms} = \sqrt{\frac{800T}{T}} = \sqrt{800} = 28.284$