ABSTRACT
Differentiation allows Displacement Velocity Acceleration Integration allows Acceleration Velocity Displacement
Whilst differentiation measures the gradient of the appropriate curve at a given instant, integration measures the area under a curve. This is demonstrated
Fig. A6.1a represents an object moving with constant velocity. Between two points in time, t1 and t2, we know from section A5 that:
(A5.1)
Where v is the object’s velocity, s2 – s1 is the change in displacement (Δs) between instants 1 and 2, and t2 – t1 is the change in time. Therefore:
s2 − s1 = v(t2 − t1) = area of the rectangle ABCD
The velocity data in Fig. A6.1b is more complex. This may be approximated by a number of smaller rectangles of width δt. The area of a single rectangle (given by v × δt) must be approximately equal to the change in displacement over the time δt. The area under the curve, and hence the change in displacement over the period from t1 to t2 can then be approximated by adding together the areas of all such rectangles between t1 and t2. This is only an approximation as the velocity is assumed to be constant during each small time interval δt, but the approximation gets closer to reality the smaller δt is, and is exact if δt = 0.
