The arc-length differential

This video demonstrates the derivation of the arc length formula using the method of infinitesimals. It begins by dividing a curve into small segments, showing through zooming that short curved segments approximate straight tangent lines. By constructing a right triangle with sides $dx$, $dy$, and hypotenuse $ds$, it applies the Pythagorean theorem to derive the differential element $ds = \sqrt{1 + (dy/dx)^2} dx$. Finally, summing these elements via integration yields the definite integral formula for arc length: $L = \int_a^b \sqrt{1 + (dy/dx)^2} dx$.

Before you watch

• Geometric meaning of derivatives (tangent slope)
• Pythagorean theorem
• Definition of definite integrals (limit of Riemann sums)