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How do you find the arc length of the curve #y=ln(sec x)# from (0,0) to #(pi/ 4,1/2ln2)#?

Answer 1

Emily Ammerman

Your graph, in the requested interval, should look like:

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Answer 2

Luke Alvarez

To find the arc length of the curve ( y = \ln(\sec x) ) from ( (0,0) ) to ( \left(\frac{\pi}{4},\frac{1}{2}\ln2\right) ), you use the formula for arc length:

[ L = \int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} , dx ]

First, find the derivative of ( y = \ln(\sec x) ).
Then, square the derivative and add 1.
Integrate this expression with respect to ( x ) from ( 0 ) to ( \frac{\pi}{4} ).

The integral setup would be:

[ L = \int_{0}^{\frac{\pi}{4}} \sqrt{1 + \left(\frac{d}{dx}\ln(\sec x)\right)^2} , dx ]

After finding the integral, evaluate it from ( 0 ) to ( \frac{\pi}{4} ) to get the arc length.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.