Arc length accumulates tiny hypotenuses along a curve.
For f(x)=x²/2, the slope is x. A short segment has length approximately √(1+x²)Δx. In the limit, the integral from 0 to b is [b√(1+b²)+asinh b]/2. The red polygon is a finite approximation, not the exact length.
Arc length is not the area under the curve.