What is the arc length of $f(x)=(3/2)x^(2/3)$ on $x in [1,8]$ ?

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Answer 1 · 2018-03-19T17:46:39

$L=5sqrt5-2sqrt2$ units.

$f(x)=3/2x^(2/3)$

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

Arc length is given by:

$L=int_1^8sqrt(1+x^(-2/3))dx$

Simplify:

$L=int_1^8sqrt(1+x^(2/3))(x^(-1/3)dx)$

Apply the substitution $x^(2/3)=u$ :

$L=3/2int_1^4sqrt(1+u)du$

Integrate directly:

$L=[(1+u)^(3/2)]_1^4$

Hence

$L=5sqrt5-2sqrt2$

What is the arc length of f(x)=(3/2)x^(2/3)f(x)=(3/2)x^(2/3) on x in [1,8]x in [1,8]?