# How do you find the arc length of the curve f(x)=2(x-1)^(3/2) over the interval [1,5]?

May 16, 2018

Use the arc length formula.

#### Explanation:

f(x)=2(x−1)^(3/2)

$f ' \left(x\right) = 3 \sqrt{x - 1}$

Arc length is given by:

$L = {\int}_{1}^{5} \sqrt{1 + 9 \left(x - 1\right)} \mathrm{dx}$

Simplify:

$L = {\int}_{1}^{5} \sqrt{9 x - 8} \mathrm{dx}$

Integrate directly:

$L = \frac{2}{27} {\left[{\left(9 x - 8\right)}^{\frac{3}{2}}\right]}_{1}^{5}$

Hence:

$L = \frac{2}{27} \left({37}^{\frac{3}{2}} - 1\right)$