A car moves along a straight road in such a way that its...

A car moves along a straight road in such a way that its velocity (in feet per second) at any time $t$ (in seconds) is given by
$$v(t)=3 t \sqrt{16-t^{2}} \quad(0 \leq t \leq 4)$$
Find the distance traveled by the car in the 4 sec from $t=0$ to $t=4$.

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Key Concepts

Definite Integration

Definite integration is used to calculate the area under a curve between fixed limits, and in the context of motion, it provides the total displacement or distance traveled over a specific time interval. Here, integrating the velocity function with respect to time yields the distance the car travels from the starting to the ending time.

Relationship Between Velocity and Displacement

In kinematics, the displacement of an object moving along a straight line is found by integrating its velocity function over the time interval of interest. This principle links the rate of change of position (velocity) with the overall change in position (displacement), forming a foundational concept in both classical mechanics and calculus.

Substitution Method in Integration

The substitution method is a technique used to simplify the evaluation of integrals, especially when dealing with composite functions such as those involving square roots. This method involves changing the variable of integration to transform the original integral into a more manageable form, which is particularly useful when integrating functions that resemble forms like sqrt(a - t^2).

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