Question

Given the following periodic function,
$f\left(t\right)=\{\begin{array}{c}{t}^{2}for0\le t\le 2\\ -t+6for2\le t\le 6\end{array}$

the co-efficient $a_0$ of the continuous fourier series associated with f(t) is given as $\frac{X}{9},$ the value of X is .

• A. 16

Answer 1

The correct option is A 16
The coefficient $a_0$ of the continuous fourier serier associated with the given function f(t) can be computed as

$a_0=\left(\frac{1}{T}\right){\int }_0^{T}f\left(t\right)dt$
$a_0=\frac{1}{6}[\underset{0}{\overset{2}{\int }}{t}^{2}dt+\underset{2}{\overset{6}{\int }}(-t+6\left)dt\right]$

$a_0=\frac{1}{6}[{\left(\frac{t^3}{3}\right)}_0^2+{(\frac{-t^2}{2}+6\times t)}_2^6]$

$a_0=1.78=\frac{16}{9}$
$\Rightarrow X=16$