A metallic rod of length L is rotated with an angular speed of ω normal to a uniform magnetic field B…

Linear velocity at the distance $x$ from the centre can be written as, $v=\omega x$.

Therefore, induced EMF due to small element will be,

$d\epsilon =Bv\mathrm{d}x=B\left(\omega x\right)\mathrm{d}x$

Integrating both sides, we get

$\int d\epsilon =\int B\left(\omega x\right)dx$

$\Rightarrow \epsilon =B\omega {\int }_0^{L}xdx=\frac{B\omega L^2}{2}$