A parabolic cable is held between two supports at the same level. The horizontal span between the supports is L. The sag at the mid span is h. The equation of the parabola is \(y=4h\frac{x^2}{L^2}\), where x is the horizontal coordinate and y is the vertical coordinate with the origin at the centre of the cable. The expression for the total length of the cable is:
1
\(\mathop \displaystyle \int \nolimits_0^{\rm{L}} \sqrt {\begin{array}{*{20}{c}} {1 + 64\dfrac{{{{\rm{h}}^2}{{\rm{x}}^2}}}{{{{\rm{L}}^4}}}}\\ \; \end{array}} {\rm{\;dx}}\)
2
\(2\mathop \displaystyle \int \nolimits_0^{{\rm{L}}/2} \sqrt {\begin{array}{*{20}{c}} {1 + 64\dfrac{{{{\rm{h}}^3}{{\rm{x}}^2}}}{{{{\rm{L}}^4}}}}\\ \; \end{array}} {\rm{\;dx}}\)
3
\(\mathop \displaystyle \int \nolimits_0^{{\rm{L}}/2} \sqrt {\begin{array}{*{20}{c}} {1 + 64\dfrac{{{{\rm{h}}^2}{{\rm{x}}^2}}}{{{{\rm{L}}^4}}}}\\ \; \end{array}} {\rm{\;dx}}\)
4
\(2\mathop \displaystyle \int \nolimits_0^{{\rm{L}}/2} \sqrt {\begin{array}{*{20}{c}} {1 + 64\dfrac{{{{\rm{h}}^2}{{\rm{x}}^2}}}{{{{\rm{L}}^4}}}}\\ \; \end{array}} {\rm{\;dx}}\)