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Question

# Write and simplify the integral that gives the arc length of the following curves on the given interval.y=$\frac { 1 } { x }$ on [1, 10]

Solution

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$\begin{gathered} y = \frac{1}{x};\left[ {\underbrace 1_a,\underbrace {10}_b} \right] \\ a) \\ \textcolor{#4257b2}{ {\text{Differentiate}}} \\ \frac{{dy}}{{dx}} = - \frac{1}{{{x^2}}} \\ \textcolor{#4257b2}{ {\text{Apply }}L = \int_a^b {\sqrt {1 + {{\left( {\frac{{dy}}{{dx}}} \right)}^2}} } dx} \\ L = \int_1^{10} {\sqrt {1 + {{\left( { - \frac{1}{{{x^2}}}} \right)}^2}} } dx \\ \textcolor{#4257b2}{{\text{Simplify}}{\text{ the integrand}}} \\ L = \int_1^{10} {\sqrt {1 + \frac{1}{{{x^4}}}} } dx \\ b) \\ \textcolor{#4257b2}{{\text{Approximate the integral using a calculator}}} \\ L = \int_1^{10} {\sqrt {1 + \frac{1}{{{x^4}}}} } dx \approx 9.1526 \\ \end{gathered}$

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