$S=\int_a^b \sqrt{1+[f'(x)]^2} dx$

The derivative of the function $f(x)$ with respect to $x$.

The arc length formula uses the Pythagorean theorem ($c=\sqrt{a^2+b^2}$) to find the length of small segments on the curve, where 1 represents the square of the length along the x-axis and $[f'(x)]^2$ represents the square of the length along the y-axis.

Arc length provides a way to calculate the distance traveled along a curve, even when the curve is complex and continuously changing.

It allows us to calculate the distance traveled along a curve, which is crucial in various applications in mathematics and physics.

1. Find $f'(x) = 2x$. 2. Apply the arc length formula: $S=\int_0^3 \sqrt{1+(2x)^2} dx$. 3. Evaluate the integral (approximately 3.823).

1. Define the function $f(x)$. 2. Find the derivative $f'(x)$. 3. Apply the arc length formula $S = \int_a^b \sqrt{1 + [f'(x)]^2} dx$. 4. Evaluate the integral.