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

preprekomW
2021-03-05
Answered

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

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FieniChoonin

Answered 2021-03-06
Author has **102** answers

Step 1

It is known that the area bounded by the curves$y=f\left(x\right){\textstyle \phantom{\rule{1em}{0ex}}}\text{and}{\textstyle \phantom{\rule{1em}{0ex}}}y=g\left(x\right)on[a,b]$ is given by

$A={\int}_{a}^{b}$ (upper curve - lower curve) dx.

From the figure, the equation of latus rectum is$y=c$ which is upper curve.

Note that the given graph is about y-axis.

Step 2

Substitute$y=c\in y=\frac{{x}^{2}}{4c}$

and obtain that$x=2x$

Thus, the area bounded can be computed as follows.

$A=2{\int}_{0}^{2c}$ (upper curve - lower curve) dx

$=2{\int}_{0}^{2c}(c-\frac{{x}^{2}}{4c})dx$

$=2{[cx-\frac{{x}^{3}}{12c}]}_{0}^{2c}$

$=2[2{c}^{2}-\frac{2}{3}{c}^{2}]$

$=\frac{8}{3}{c}^{2}$

It is known that the area bounded by the curves

From the figure, the equation of latus rectum is

Note that the given graph is about y-axis.

Step 2

Substitute

and obtain that

Thus, the area bounded can be computed as follows.

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