$($a$)$ A simply supported beam carries a point load $P$ at the mid span

as shown in the following figure.

The deflection $y,$ along the beam is governed by the

following equation:

$y=\frac{Px}{48EI}(-4x^2+3L^2)$ for $0x\frac{L}{2}$

where

$x$ is the distance from the support $A$;

$L$ is the total length of the beam;

$P$ is the point load acting at the mid span;

$E$ is the Young's modulus of the beam, and

I is the moment of inertia of the section

of beam.

$($Hint: $L,P,E$ and I are constant parameters$)$

$($i$)$ By using the application of differentiation, find the

value of $x$ where the deflection $y$ is maximum;

$($ii$)$ Find the maximum deflection, give your answer in terms of

$L,P,E$ and I.

$($b$)$ Find the area bounded by the curve $y=\frac{Px}{48EI}(-4x^2+3L^2)$

$(10$ marks$)$

and the $x-$axis from $x=0$ to $x=\frac{L}{2},$ give your answer in

terms of $L,P,E$ and I.