How do you find the lengths of the curve $y=(x-1)^(2/3)$ for $1<=x<=9$ ?

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Answer 1 · 2018-03-23T10:08:06

Work in terms of $y$ .

$y=(x−1)^(2/3)$ with $1≤x≤9$

Invert:

$x=y^(3/2)+1$ with $0≤y≤4$

$x'=3/2sqrty$

Arc length is given by:

$L=int_0^4sqrt(1+9/4y)dy$

Integrate directly:

$L=8/27[(1+9/4y)^(3/2)]_0^4$

Hence

$L=8/27(10sqrt10-1)$

How do you find the lengths of the curve y=(x-1)^(2/3)y=(x-1)^(2/3) for 1<=x<=91<=x<=9?