Final answer:
The covered displacement of the function v = t³ - 3t² + 2t from time t = 0 to t = 3 is found by integrating the velocity function over this interval, resulting in a displacement of 2.25 length units.
Step-by-step explanation:
To find the covered displacement of the function v = t³ - 3t² + 2t within the time interval 0 to 3, we need to determine the definite integral of the velocity function with respect to time over that interval. The displacement is the area under the velocity-time graph between these two time points.
The integral of the velocity function v(t) from 0 to 3 would be:
≈(t) = ∫ (t³ - 3t² + 2t) dt
Find the antiderivative: ≈(t) = ⅔t⁴ - t³ + t² + C
Now, evaluate this antiderivative from 0 to 3: [⅔(3)⁴ - (3)³ + (3)²] - [⅔(0)⁴ - (0)³ + (0)²]
≈(t) = [⅔×¹¹ - 27 + 9] - [0]
This simplifies to: ≈(t) = ⅔×¹¹ - 18
Calculate the value of the antiderivative at the limits: 20.25 - 18 = 2.25
Therefore, the covered displacement from t = 0 to t = 3 is 2.25 length units.