# How do you find the arc length of the curve y=sqrt(x-3) over the interval [3,10]?

Mar 13, 2018

Arc length will be approximately $7.716$ units.

#### Explanation:

Recall that arc length of a curve is given by

$A = {\int}_{a}^{b} \sqrt{1 + {\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^{2}} \mathrm{dx}$

The derivative of our curve is given by the chain rule as being $\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{1}{2 \sqrt{x - 3}}$.

$A = {\int}_{3}^{10} \sqrt{1 + {\left(\frac{1}{2 \sqrt{x - 3}}\right)}^{2}} \mathrm{dx}$

$A = {\int}_{3}^{10} \sqrt{1 + \frac{1}{4 \left(x - 3\right)}} \mathrm{dx}$

$A = {\int}_{3}^{10} \sqrt{\frac{4 x - 12 + 1}{4 \left(x - 3\right)}} \mathrm{dx}$

$A = {\int}_{3}^{10} \sqrt{\frac{4 x - 11}{4 x - 12}} \mathrm{dx}$

This is a pretty complex integral, so I would solve using a graphing calculator.

Evaluating you should get $A = 7.716$ units.

Hopefully this helps!