# Length of Curve with multivariable

• Jul 8th 2009, 05:49 PM
sleepiiee
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?
• Jul 8th 2009, 06:08 PM
Random Variable
$\displaystyle \text{length} = \int^{\pi /4}_{0} ||r'(t)|| \ dt$

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

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

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

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

Can you integrate that?
• Jul 8th 2009, 06:19 PM
sleepiiee
how did you derive log(cost) to be -tant

did you just change it to ln?
• Jul 8th 2009, 06:21 PM
Random Variable
Quote:

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?
• Jul 8th 2009, 06:53 PM
sleepiiee
its blank, would that mean its base is 10 or e?
• Jul 8th 2009, 07:02 PM
Random Variable
Quote:

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

most likely base e
• Jul 8th 2009, 08:20 PM
sleepiiee
i got ln|secx + tanx| evaluated at 0 and pi/4

and got the answer ln|(2)^1/2|

can anyone confirm that the answer is correct?
• Jul 8th 2009, 08:25 PM
sleepiiee
nvm i got ln[(rad 2) + 1]
• Jul 8th 2009, 08:28 PM
Random Variable
ln |secx +tanx| is the antiderivative of secx

since the derivative of ln(cosx) is -tanx, the antiderivative of tanx is -ln(cosx)