Question

What is the arc length of `f(x)=secx*tanx` in the interval `[0,pi/4]`?

Answer 1

The arc length formula is basically the distance formula adapted to a function. Whereas the distance formula would be `D = sqrt((Deltax)^2 + (Deltay)^2)`, the arc length formula is:

`\mathbf(s = int_a^b sqrt(1 + ((dy)/(dx))^2)dx)`

So, we need to take the derivative and then square it.

`d/(dx)[secxtanx]`

`= secx*sec^2x + secxtan^2x`

`= secx(sec^2x + tan^2x)`

`= secx(1 + 2tan^2x)`

And squaring it...

`((dy)/(dx))^2`

`= color(green)(sec^2x(2tan^2x + 1)^2)`

And now putting it in the integral.

`s = color(blue)(int_0^(pi"/"4) sqrt(1 + sec^2x(2tan^2x + 1)^2) dx)`

At that point, you should evaluate the integral numerically on your calculator. A TI-89 should be able to do it, but working it out would be far too complicated.

I get `~~ color(blue)(1.6458)` using Wolfram Alpha, and this integral returns the same answer.

As you can see, the setup is usually pretty easy. What's usually hard (and generally nearly impossible on medium-complexity functions) is the actual simplification work.