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

\(\displaystyle={2}{\int_{{0}}^{{{2}{c}}}}{\left({c}-\frac{{x}^{2}}{{{4}{c}}}\right)}{\left.{d}{x}\right.}\)

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

\(\displaystyle={2}{\left[{2}{c}^{2}-\frac{2}{{3}}{c}^{2}\right]}\)

\(\displaystyle=\frac{8}{{3}}{c}^{2}\)

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

\(\displaystyle={2}{\int_{{0}}^{{{2}{c}}}}{\left({c}-\frac{{x}^{2}}{{{4}{c}}}\right)}{\left.{d}{x}\right.}\)

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

\(\displaystyle={2}{\left[{2}{c}^{2}-\frac{2}{{3}}{c}^{2}\right]}\)

\(\displaystyle=\frac{8}{{3}}{c}^{2}\)