What is the arc length of $f (x) = \frac{3 x}{\sqrt{x - 1}}$ $f(x)=(3x)/sqrt(x-1)$ on $x \in [2, 6]$ $x in [2,6]$ ?

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Answer 1 · 2018-07-08T18:45:29

$\approx 4.514518184$ $approx 4.514518184$

Writing

$f (x) = 3 x {(x - 1)}^{- \frac{1}{2}}$ $f(x)=3x(x-1)^(-1/2)$
then by the product and the chain rule we get

$f'(x)=3(x-1)^(-1/2)+3x*(-1/2)*(x-1)^(-3/2)$

which simplifies to

$f'(x)=(3(x-2))/(2*(x-1)^(3/2))$
so we have to solve

$int_2^6 sqrt(1+(3(x-2)/(2*(x-1)^(3/2)))^2)dx$
by a numerical method we get

$\approx4.514518184$

What is the arc length of f(x)=(3x)/sqrt(x-1) f(x)=3x√x−1f(x)=(3x)/sqrt(x-1) on x in [2,6] x∈[2,6]x in [2,6] ?