Question

What curve passes through the point ​(1 ​,2​) and has an arc length on the interval​ [2,6] given by Integral from 2 to 6 StartRoot 1 plus 64 x Superscript negative 6 EndRoot dx ​? What is the​ curve?

Answer 1

Explanation:

Given

Length of curve

`L=\int_(2)^(6)\sqrt{1+64x^(-6)}dx`

Length of curve is given by

`L=\int_(a)^(b)\sqrt{1+\left ( \frac{\mathrm{d} y}{\mathrm{d} x}\right )^2}dx` over interval a to b

comparing two we get

`\frac{\mathrm{d} y}{\mathrm{d} x}=8x^(-3)`

`dy=8x^(-3)dx`

integrating

`\int dy=\int 8x^(-3)dx`

`y=-4x^(-2)+C`

Curve Passes through (1,2)

`1=-4+C`

`C=5`

curve is

`y+(4)/(x^2)=5`