Compute the arc length of y = ln(sec x), 0 less than or equal to x less than or equal to pi/4.
Question:
Compute the arc length of {eq}\; y = \ln(\sec x), \; 0 \leq x \leq \frac{\pi}{4} {/eq}.
Finding the Arc Length:
To solve this problem, we'll use the formula for a smooth curve.
The formula for the length of {eq}y=f\left ( x \right ), a\leq x\leq b {/eq}:
If {eq}f{}' {/eq} is continuous on {eq}\left [ a,b \right ] {/eq}, the length of the curve {eq}y=f\left ( x \right ) {/eq} from {eq}A=\left ( a,f\left ( a \right ) \right )\;\text{to }B=\left (b, f\left ( b \right )\right ) {/eq} is given by {eq}\int_{a}^{b}\sqrt{1+\left ( \frac{\mathrm{d}y}{\mathrm{d}x} \right )^{2}}\mathrm{d}x {/eq}.
Answer and Explanation:
Given:
{eq}y=\ln\left ( \sec x \right ) {/eq}
Differentiating with respect to {eq}x {/eq}, we get
{eq}\begin{align*} \frac{\mathrm{d}...
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