Question

9–20. Arc length calculations Find the arc length of the following curves on the given interval.y = ln (x−√x²−1), for 1 ≤ x ≤ √2(Hint: Integrate with respect to y.)​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​

Accepted Answer

Recognize that the problem asks for the arc length of the curve defined by $$y = \ln \left(x - \sqrt{x^2$ - 1}\right) on $$the interval $$1 \leq x \leq \sqrt{2}. $$The hint suggests integrating with respect to $$y$$, so consider expressing $$x as a $$function of $$y$$ instead of $$y as a $$function of $$x. $$Start by rewriting the given function to find $$x in $$terms of $$y. $$Since $$y = \ln \left(x - \sqrt{x^2$ - 1}\right)$$, exponentiate both sides to get $$e^y = x - \sqrt{x^2$ - 1}. $$Then solve this equation for $$x. $$After isolating $$x$$, differentiate $$x$$ with respect to $$y to $$find $$\frac{dx}{dy}. $$This derivative will be used in the arc length formula when integrating with respect to $$y. $$Recall the arc length formula when integrating with respect to $$y$$: $$L = \int_{y_1$}^{y_2$} \sqrt{1 + \left(\frac{dx}{dy}\$$right)^2$$} \, dy. $$Determine the new limits of integration $$y_1$$ and $$y_2 by $$substituting the original $$x-$$values into the original function to find corresponding $$y-$$values. Set up the integral for the arc length using the expression for $$\frac{dx}{dy}$$ and the limits $$y_1$$ and $$y_2. $$The final step is to evaluate this integral to find the arc length, but per instructions, do not compute the final value here.

9–20. Arc length calculations Find the arc length of the following curves on the given interval.
y = ln (x−√x²−1), for 1 ≤ x ≤ √2(Hint: Integrate with respect to y.)

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Key Concepts

Arc Length Formula

Inverse Hyperbolic Functions and Logarithms

Integration with Respect to y