Vault Of Euclid

Distance Traveled by a Particle with Cubic Velocity

Antiderivatives core

A particle moves on a straight line with velocity \( v = (4-2t)^3 \) at time \( t \). Find the distance traveled from \( t=0 \) to \( t=3 \).

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What you need first

Integration by substitution
Understanding the relationship between position and velocity
Evaluating absolute values

What this problem tests

Tests the ability to integrate using substitution and to correctly handle the absolute value of velocity by splitting the interval at turning points.

This is a standard application of integration in introductory calculus, testing the distinction between net displacement and total distance traveled.

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