A cyclist begins moving along a straight road at time t=0 seconds. For the next 12 seconds the velocity v(t) of the cyclist (in meters per second) is known to be a continuous, differentiable function.
The total displacement of the cyclist from t=0 to t=12 seconds is represented by the definite integral ∫012v(t)dt.
(a) Describe, in complete sentences, the physical meaning of this definite integral in the context of the motion of the cyclist.
(b) The quantity 121∫012v(t)dt is another characteristic of the motion. Explain what this quantity measures, include the appropriate units, and discuss how it differs conceptually from the displacement calculated in part (a).
(c) Suppose that v(t) changes sign during the interval 0≤t≤12. Explain how this sign change would affect your interpretation in parts (a) and (b), and clarify whether the quantity in part (b) can ever be negative.
(a) Describe the physical meaning of ∫012v(t)dt.
(b) Explain what 121∫012v(t)dt measures and include its units.
(c) Discuss the effect of v(t) changing sign within the interval on the interpretations from parts (a) and (b).