Find the total distance traveled by a particle whose position is given by parametric equations involving squared trigonometric functions.

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Distance Traveled by a Particle on a Parametric Curve

Parametric equations and vector functions core

Find the distance traveled by a particle with position \((x, y)\) as \(t\) varies in the given time interval: \(x = \sin^2 t, y = \cos^2 t, 0 \le t \le 3\pi\).

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What you need first

Derivatives of trigonometric functions
The chain rule
Double-angle identities for sine
Evaluating definite integrals
Understanding the difference between distance traveled and displacement

What this problem tests

Tests the student's ability to apply the parametric arc length formula, simplify trigonometric expressions, and correctly handle the integration of an absolute value function by exploiting its periodicity.

This problem typically appears in a Calculus II or Multivariable Calculus course when introducing calculus with parametric equations.

Distance Traveled by a Particle on a Parametric Curve

What you need first

What this problem tests

More on Parametric equations and vector functions