Single Variable Calculus: Arc length

We look at how to compute the arc length for differentiable parametric curves r(t) = (x(t), y(t)), a ≤ t ≤ b, in the xy-plane. This includes computing the lengths of curves which are pieces of the graphs of functions of the form y=f(x) and x=g(y). Examples provided range from straightforward integrations to those requiring setup without actual integration due to complexity. Additionally, we go through a derivation of the arc length formula (using a Riemann sum). Key points: - Arc Length Formula: Understand the derivation and application of the integral formula used to calculate the length of a curve described parametrically. - Differentiation and Integration: Learn how to differentiate parametric equations, square the derivatives, and set up the integral necessary to find the arc length. - Practical Examples: Through various examples, see how to apply the formula in different scenarios, including cases where the integral is not straightforward to compute. - Formula Derivation: Gain insights into the theoretical underpinning of the arc length formula, using a Riemann sum. 00:00 Intro 01:22 First Example: Arc Length of a Parametric Curve 08:58 Second Example: Finding Arc Length Using Parametric Equations 12:42 Third Example: Setting up Arc Length for x = 2y^2 15:52 Fourth Example: Computing Arc Length for r(t) = (t^2, t^3+1) 27:24: Fifth Example: Verifying the Arc Length for a circle 34:07 Derivation of the Arc Length Formula #mathematics #math #calculus #integralcalculus #calculus2 #arclength #parametriccurves #mathtutorial #integralcalculus #integration #applicationsofintegration