Find a curve that passes through the point $(1,3)$ and has an arc length on the interval $[2,6]$ given by $\int_{2}^{6} \sqrt{1+16 x^{-6}} d x$ ?

Step 1 ：Given the arc length of a curve is represented by the integral \(\int_a^b \sqrt{1+(\frac{dy}{dx})^2} dx\), we can compare this with the given integral \(\int_{2}^{6} \sqrt{1+16 x^{-6}} dx\) to find that \(\frac{dy}{dx} = 4x^{-3}\).

Step 2 ：We can integrate \(\frac{dy}{dx} = 4x^{-3}\) to find the function y(x). The integral of \(4x^{-3}\) is \(-2x^{-2} + C\), where C is the constant of integration.

Step 3 ：We know that the curve passes through the point (1,3). Substituting these values into the equation \(y = -2x^{-2} + C\), we get \(3 = -2(1)^{-2} + C\). Solving for C, we find that C = 5.

Step 4 ：Substituting C = 5 back into the equation \(y = -2x^{-2} + C\), we get the final equation of the curve as \(y = 5 - \frac{2}{x^2}\).