Find the average position of an object over a time interval given its velocity function and initial position.

Vault Of Euclid

Average Position of an Object with Cosine Velocity

The definite integral core

The velocity \( v \) of an object moving on the \( x \)-axis is \( \cos 3t \), and the object is at the origin at \( t=0 \). Find the average value of the position \( x \) over the interval \( 0 \le t \le \pi/3 \).

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What you need first

Integration of trigonometric functions
Solving initial value problems for a constant of integration
Evaluating definite integrals
The formula for the average value of a function

What this problem tests

Tests the ability to integrate a trigonometric function, solve for the constant of integration using an initial condition, and apply the average value formula.

This is a standard application of the definite integral in a first-year calculus course, combining kinematics with the average value theorem.

Average Position of an Object with Cosine Velocity

What you need first

What this problem tests

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