Arc Length Calculator for Curve

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

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Solution

Your input: find the arc length of $$$f\left(x\right)=\sqrt{x}$$$ on $$$\left[0,2\right]$$$.

The length of the curve is given by $$$L = \int_a^b \sqrt{\left(f'\left(x\right)\right)^2+1}d x$$$.

First, find the derivative: $$$f '\left(x\right)=\left(\sqrt{x}\right)'=\frac{1}{2 \sqrt{x}}$$$ (steps can be seen here)

Finally, calculate the integral $$$L = \int_{0}^{2} \sqrt{\left(\frac{1}{2 \sqrt{x}}\right)^{2} + 1} d x=\int_{0}^{2} \frac{\sqrt{4 + \frac{1}{x}}}{2} d x$$$

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