Suppose a particle moves back and forth along a straight line with velocity \(v(t)\), measured in feet per second, and acceleration \(a(t)\). What are the meanings of the following \(\displaystyle\int_{0}^{30}v(t)dt\), \(\displaystyle\int_{0}^{30}|v(t)|dt\), and \(\displaystyle v(t)|_{0}^{30}\).

The terms used here are the following:

Displacement: of the particle measured in feet after travelling from 0 to 30 seconds.

Change in speed: of the particle measured in feet per seconds.

Distance: Total distance travelled by the particle in feet.

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