Arc Length from a Parametric Curve?

📌 CONCEPT: The arc length of a parametric curve can be calculated using the formula derived from the Pythagorean theorem, which involves integrating the square root of the sum of the squares of the derivatives of the parametric equations with respect to the parameter t.

📐 RULE / FORMULA: The formula for the arc length of a parametric curve given by x = f(t) and y = g(t) from t = a to t = b is ∫[a,b] √((dx/dt)^2 + (dy/dt)^2)dt.

💡 WORKED EXAMPLE: For the given parametric curve x = 2cos(t) and y = 3sin(t), the derivatives are dx/dt = -2sin(t) and dy/dt = 3cos(t). Substituting these into the arc length formula, we get ∫[0,π/2] √((-2sin(t))^2 + (3cos(t))^2)dt. Evaluating this integral gives us the arc length covered by the bug in the first 5 seconds.

⚠️ COMMON MISTAKE: Students often forget to square the derivatives before substituting them into the arc length formula, leading to incorrect calculations.