Question

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.

Conic sections
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.

2021-03-06
Step 1
It is known that the area bounded by the curves $$\displaystyle{y}= f{{\left({x}\right)}}{\quad\text{and}\quad}{y}= g{{\left({x}\right)}}{o}{n}{\left[{a},{b}\right]}$$ 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 $$\displaystyle{y}={c}\in{y}=\frac{{x}^{2}}{{{4}{c}}}$$
and obtain that $$\displaystyle{x}={2}{x}$$
Thus, the area bounded can be computed as follows.
$$\displaystyle{A}={2}{\int_{{0}}^{{{2}{c}}}}$$ (upper curve - lower curve) dx
$$\displaystyle={2}{\int_{{0}}^{{{2}{c}}}}{\left({c}-\frac{{x}^{2}}{{{4}{c}}}\right)}{\left.{d}{x}\right.}$$
$$\displaystyle={2}{{\left[{c}{x}-\frac{{x}^{3}}{{{12}{c}}}\right]}_{{0}}^{{{2}{c}}}}$$
$$\displaystyle={2}{\left[{2}{c}^{2}-\frac{2}{{3}}{c}^{2}\right]}$$
$$\displaystyle=\frac{8}{{3}}{c}^{2}$$