14
Integration
When we first encountered the concept of derivatives in Chapter 12, we introduced it through an example from physics. As Newton created it, the derivative describes the velocity of a moving object as calculated from its time-distance graph. In other words, the velocity can be derived from the time-distance information.
Can the distance be reconstructed given the velocity? In a sense, this is the inverse of differentiation.
Questions such as these are hard to answer if we only look at the most general case, so let’s consider a special one. Suppose that our object is moving with a constant velocity v(t) = v0
, for a duration of T seconds. With some elementary logic, we can conclude that the total distance traveled is v0T meters.
When taking a look at the time-velocity plot, we can immediately see that the distance is the area under the time-velocity function graph v(t) = v0.
The graph of v(t) describes a rectangle with width v0 and length T, hence its area is indeed...
