A latus rectum of a conic section is a chord through a focus parallel to the directrix. Find the area bounded by the parabola displaystyle{y}={x}^{2}text{/}{left({4}{c}right)} and its latus rectum.

Step 1
It is known that the area bounded by the curves ${\displaystyle y=f\left(x\right)\text{and}y=g\left(x\right)on[a,b]}$ is given by
${\displaystyle A={\int }_a^b}$ (upper curve - lower curve) dx.
From the figure, the equation of latus rectum is ${\displaystyle y=c}$ which is upper curve.
Note that the given graph is about y-axis.
Step 2
Substitute ${\displaystyle y=c\in y=\frac{x^2}{4c}}$
and obtain that ${\displaystyle x=2x}$
Thus, the area bounded can be computed as follows.
${\displaystyle A=2{\int }_0^{2c}}$ (upper curve - lower curve) dx
${\displaystyle =2{\int }_0^{2c}(c-\frac{x^2}{4c})dx}$
${\displaystyle =2{[cx-\frac{x^3}{12c}]}_0^{2c}}$
${\displaystyle =2[2c^2-\frac{2}{3}{c}^2]}$
${\displaystyle =\frac{8}{3}{c}^2}$