# How do you find the arc length of the curve y = 4x^(3/2) - 1 from [4,9]?

Jun 22, 2015

Arc length would be ${\int}_{4}^{9} \left(\sqrt{1 + {\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^{2}}\right) \mathrm{dx}$

= ${\int}_{4}^{9} \left(\sqrt{1 + 36 x}\right) \mathrm{dx}$

=${\left[\frac{2}{3} \cdot \frac{1}{36} {\left(1 + 36 x\right)}^{\frac{3}{2}}\right]}_{4}^{9}$

=$\frac{1}{54} \left[{\left(343\right)}^{\frac{3}{2}} - {\left(145\right)}^{\frac{3}{2}}\right]$

= $\frac{1}{54} \left[{7}^{\frac{9}{2}} - {\left(145\right)}^{\frac{3}{2}}\right]$
This can be simplified, if required, to get an approximation, using calculator.

Jun 23, 2015

76.1664

#### Explanation:

An error has crept into the solution. The correction is as follows:

Arc length= $\frac{1}{54} {\left[{\left(1 + 36 x\right)}^{\frac{3}{2}}\right]}_{4}^{9}$

= $\frac{1}{54} \left[{325}^{\frac{3}{2}} - {145}^{\frac{3}{2}}\right]$

This can be simplified, if required as $\frac{1}{54} \left[5859.020 - 1746.031\right]$
= 76.1664