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.