Arc Length

Q. 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. ${\int }_0^L\sqrt{1+64\frac{h^2{x}^2}{L^4}dx}$
2. $2{\int }_0^{L/2}\sqrt{1+64\frac{h^3{x}^2}{L^4}dx}$
3. ${\int }_0^{L/2}\sqrt{1+64\frac{h^2{x}^2}{L^4}dx}$
4. $2{\int }_0^{L/2}\sqrt{1+64\frac{h^2{x}^2}{L^4}dx}$

Q. $\text{The total length of the curve}r=a{\sin }^3\left(\frac{\theta }{3}\right)\text{is}ab\pi \text{. The value of b is \_\_\_\_\_ (Curve is defined for}0\le \theta \le 3\pi )$

1. 1.5