Question

Find the equation of the curve that passes through the point (x, y) = (0, 0) and has an arc length on the interval x is between 0 and pi over 4 inclusive given by the integral the integral from 0 to pi over 4 of the square root of the quantity 1 plus the secant to the 4th power of x, dx .

Answer 1

$\displaystyle\int_0^{\pi/4}\sqrt{1+\sec^4x}\,\mathrm dx=\int_0^{\pi/4}\sqrt{1+(\sec^2x)^2}\,\mathrm dx$

Recall that $\displaystyle(\tan x)'=\sec^2x$. Then right away you see the integral gives the arc length of the curve $y=\tan x$ over the given interval.