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Arclength of the curve $y= \ln( \sec x)$ $ 0 \le x \le \pi/4$

I know that I have to find its derivative which is easy, it is $\tan x$

Then I put it into the arclength formula

$\int \sqrt {1 - \tan^2 x}$

From here I am not sure what to do, I put it in wolfram and it got something massive looking. I know I can't use u substitution and I am pretty certain I have to algebraicly manipulate this before I can continue but I do not know how.

  • 1
    @Jordan You don't need to know the trig - remember that the arc length formula _in general_ doesn't have to have trig in it. If the formula had been $\int \sqrt{1-\left(y'\right)^2}$, (instead of the same formula with a plus sign, as many have pointed out) then anywhere that $y'\gt 1$ you'll run into a negative square root.2012-06-11

4 Answers 4

3

The arclength formula should be $\int \sqrt{1+\tan^2 x}\ dx$.

5

You made a mistake. Arc length is given by the integral:

$ \ell = \int_a^b \sqrt{1+\left(y'\right)^2}\,dx $

So if $y' = \tan x$, arc length is:

$ \ell = \int_a^b \sqrt{1+\tan^2 x}\,dx = \int_a^b \sqrt{\sec^2 x}\,dx = \int_a^b |\sec x| \,dx $

For $a = 0$, $b = \frac{\pi}{4}$:

$ \ell = \int_0^{\pi/4} \sec x \,dx $

The absolute value is gone as $\sec x$ is positive in $[0, \frac{\pi}{4}]$.

4

The arclength formula is

$\mathrm S_a^b(f) =\int_a^b \sqrt{1+f'(x)^2}dx$

You have

$f(x) = \log \sec x$

This means

$f'(x) = \tan x$

Then you need to find

$\mathrm S =\int_0^{\pi/4} \sqrt{1+\tan^2 x}dx$

Remember that

$1+\tan^2 x=\sec ^2 x$

Also, remember the secant is positive in the first quadrant, so

$\mathrm S =\int_0^{\pi/4} \sqrt{\sec^2 x}dx$

$\mathrm S =\int_0^{\pi/4} \sec xdx$

4

As others have noted, it should be

$\int_0^{\pi/4} \sqrt{1+\tan^2 x} \ \ dx$ Recall $1+\tan^2 x = \sec^2 x$ $\int_0^{\pi/4} \sqrt{\sec^2 x} \ \ dx$ Since all trig functions are positive in the first quadrant, we can simply rewrite the integrand as $\int_0^{\pi/4} {\sec x} \ \ dx$

Which is a (relatively) well known integral that evaluates to $\log (\tan(x) + \sec (x))$

Now simply evaluate at the endpoints - you should get around $.8814$ assuming I didn't make a button-punching error.