int_0^{1/2} |sin(4pi x)| , dx = | vedclass

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(C) Let $I = \int_0^{1/2} |\sin(4\pi x)| \, dx$. Since the period of $|\sin(4\pi x)|$ is $\frac{\pi}{4\pi} = \frac{1}{4}$, we can use the property of

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$\frac{1}{\pi}$

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(C) Let $I = \int_0^{1/2} |\sin(4\pi x)| \, dx$. Since the period of $|\sin(4\pi x)|$ is $\frac{\pi}{4\pi} = \frac{1}{4}$, we can use the property of periodic functions. $I = 2 \int_0^{1/4} |\sin(4\pi x)| \, dx$. In the interval $[0, 1/4]$, $\sin(4\pi x) \ge 0$, so $|\sin(4\pi x)| = \sin(4\pi x)$. $I = 2 \int_0^{1/4} \sin(4\pi x) \, dx$. $I = 2 \left[ -\frac{\cos(4\pi x)}{4\pi} \right]_0^{1/4}$. $I = -\frac{1}{2\pi} [\cos(\pi) - \cos(0)]$. $I = -\frac{1}{2\pi} [-1 - 1] = -\frac{1}{2\pi} (-2) = \frac{1}{\pi}$.

$\int_0^{1/2} |\sin(4\pi x)| \, dx =$

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