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)#