Fourier Coefficient $a_0$ Calculation

The Fourier coefficient $a_0$ represents the average value of the function $f(x)$ over the interval $[-\pi, \pi]$. It is calculated using the formula:

$a_0 = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(x) dx$

Integration Steps

The function $f(x)$ is defined piecewise. To calculate the integral $\int_{-\pi}^{\pi} f(x) dx$, we split the integral according to the function's definition:

• From $-\pi$ to $-\frac{\pi}{2}$, $f(x) = \sin 2x$.
• From $-\frac{\pi}{2}$ to $0$, $f(x) = 0$.
• From $0$ to $\pi$, $f(x) = \sin 2x$.

The total integral becomes:

$ \int_{-\pi}^{\pi} f(x) dx = \int_{-\pi}^{-\pi/2} \sin(2x) \, dx + \int_{-\pi/2}^{0} 0 \, dx + \int_{0}^{\pi} \sin(2x) \, dx $

Integral Calculation Details

1. Integral over $(-\pi, -\pi/2)$:

   $ \int_{-\pi}^{-\pi/2} \sin(2x) \, dx = \left[ -\frac{\cos(2x)}{2} \right]_{-\pi}^{-\pi/2} $

   Evaluate at the limits:

   $ = \left(-\frac{\cos(2 \times (-\pi/2))}{2}\right) - \left(-\frac{\cos(2 \times (-\pi))}{2}\right) $

   $ = \left(-\frac{\cos(-\pi)}{2}\right) - \left(-\frac{\cos(-2\pi)}{2}\right) = \left(-\frac{-1}{2}\right) - \left(-\frac{1}{2}\right) = \frac{1}{2} + \frac{1}{2} = 1 $

2. Integral over $[-\pi/2, 0]$:

   The function is $0$ in this interval:

   $ \int_{-\pi/2}^{0} 0 \, dx = 0 $

3. Integral over $(0, \pi]$:

   $ \int_{0}^{\pi} \sin(2x) \, dx = \left[ -\frac{\cos(2x)}{2} \right]_{0}^{\pi} $

   Evaluate at the limits:

   $ = \left(-\frac{\cos(2\pi)}{2}\right) - \left(-\frac{\cos(2 \times 0)}{2}\right) $

   $ = \left(-\frac{1}{2}\right) - \left(-\frac{1}{2}\right) = -\frac{1}{2} + \frac{1}{2} = 0 $

Integral Value

Sum the results from the individual intervals:

$ \int_{-\pi}^{\pi} f(x) dx = 1 + 0 + 0 = 1 $

$a_0$ Calculation

Now substitute the value of the total integral back into the formula for $a_0$:

$ a_0 = \frac{1}{2\pi} \times (1) = \frac{1}{2\pi} $

Result

The value of the Fourier coefficient $a_0$ is $\frac{1}{2\pi}$.