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.

Answer 1

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
\(\displaystyle={2}{\int_{{0}}^{{{2}{c}}}}{\left({c}-\frac{{x}^{2}}{{{4}{c}}}\right)}{\left.{d}{x}\right.}\)
\(\displaystyle={2}{{\left[{c}{x}-\frac{{x}^{3}}{{{12}{c}}}\right]}_{{0}}^{{{2}{c}}}}\)
\(\displaystyle={2}{\left[{2}{c}^{2}-\frac{2}{{3}}{c}^{2}\right]}\)
\(\displaystyle=\frac{8}{{3}}{c}^{2}\)