# Math Help - Length of Curve with multivariable

1. ## Length of Curve with multivariable

I tried to evaluate the length of this curve but cannot figure out what to do with the derivative of the log(cost). The question is:

Find the length of the curve r(t) = <1, log(cost)>
t having the domain [0, pi/4]

I did the integral from 0 to pi/4 of the square root of the sum of the derivative of the components and im having trouble integrating it. Can anyone help me solve this?

2. $\text{length} = \int^{\pi /4}_{0} ||r'(t)|| \ dt$

$r'(t) = \Big(0, \text{-}\tan t\Big)$

$length = \int^{\pi /4}_{0} \sqrt{0 + \tan^2t} \ dt$

and since $\tan t \ge 0 \ \text{for} \ 0 \le t \le \pi /4$

$= \int^{\pi /4}_{0} \tan t \ dt$

Can you integrate that?

3. how did you derive log(cost) to be -tant

did you just change it to ln?

4. Originally Posted by sleepiiee
how did you derive log(cost) to be -tant

did you just change it to ln?
I assumed it was log base e. Is it log base 10?

5. its blank, would that mean its base is 10 or e?

6. Originally Posted by sleepiiee
its blank, would that mean its base is 10 or e?
most likely base e

7. i got ln|secx + tanx| evaluated at 0 and pi/4