Length of Curve Example
\[y=f(x)\]
the length of the curve between the points \[(x_1,y_1)\]
and \[x_2,y_2\]
is \[L= \int^{x_2}_{x_1} \sqrt{1+ (\frac{dy}{dx})^2}dx\]
or \[L= \int^{y_2}_{y_1} \sqrt{1+ (\frac{dx}{dy})^2}dy\]
.Example: If
\[y^2=-4x\]
find the length of curve between \[(0,0)\]
to \[(-4,4)\]
.Differentiate
\[y^2=-4x \rightarrow \frac{dx}{dy} = - \frac{y}{2}\]
.\[\begin{equation} \begin{aligned} L &= \int^4_0 \sqrt{1+ ( \frac{-y}{2})^2} dy \\ &= [ \frac{y}{2} \sqrt{1+ \frac{y^2}{4}} + ln(y+ 2 \sqrt{1+ \frac{y^2}{4}} ]^4_0 \\ &= 2 \sqrt{5} +ln (2+ \sqrt{5} ) \end{aligned} \end{equation}\]
