# Curve Arc Length Calculator

## Calculate the arc length of a curve step by step

The calculator will try to find the arc length of the explicit, polar, or parametric curve on the given interval, with steps shown.

### Your Input

**Find the exact length of $$$y = \sqrt{x}$$$ on $$$\left[0, 2\right]$$$.**

### Solution

The length of the explicit curve is given by $$$L = \int\limits_{a}^{b} \sqrt{1+\left(f'\left(x\right)\right)^2}\, dx$$$.

First, find the derivative: $$$f'\left(x\right)=\left(\sqrt{x}\right)' = \frac{1}{2 \sqrt{x}}$$$ (for steps, see derivative calculator).

Finally, calculate the integral: $$$L = \int\limits_{0}^{2} \sqrt{1 + \left(\frac{1}{2 \sqrt{x}}\right)^{2}}\, dx = \int\limits_{0}^{2} \frac{\sqrt{4 + \frac{1}{x}}}{2}\, dx.$$$

The calculations and the answer for the integral can be seen here.

### Answer

**The calculations and the answer for the integral can be seen here.**