Vault Of Euclid

Distance Traveled with Trigonometric Velocity

The definite integral core

Find the distance traveled by an object moving along a line with velocity \( v = \sin \pi t \) from \( t = \frac{1}{2} \) to \( t = 2 \).

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What you need first

the difference between displacement and distance traveled
evaluating definite integrals
antiderivatives of trigonometric functions
the unit circle values for sine and cosine

What this problem tests

Tests the ability to distinguish between displacement and distance traveled, and to correctly split a definite integral involving an absolute value.

This problem is typical in a first-year calculus course when introducing the applications of the definite integral to rectilinear motion.

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