Differentiation and Integration::

Differentiation represents the rate of change of a function.  Integration represents an accumulation or sum of a function over a range.  They are kind of inverses of each other.

Let’s consider an example.  If you have a tank of water with a small hole at the bottom, the water would leak out and the water level would drop in the tank.  It should be obvious from common experience, that when the tank is full, the water will spurt out faster than when the water level is very low when the water flow is just a trickle.  The amount of water leaking out each second is a rate of change, that is the differentiation of the amount of water in the tank with respect to time, so when the water level is high we have a high rate of change and when its low we have a low rate of change - we write this mathematically as dV/dt.

Now, suppose we had a bucket under the hole and we collect the leaked water.  Obviously, the amount of water in the bucket will increase over time.  How would we estimate how much water is falling into the bucket over time?  Well, this isn’t so easy. Let’s say in the first second, there’s a quart of water in the bucket, but that’s not to say there will be two quarts in two seconds since the flow of water will be a little less in the second second of our experiment.  So we need to adjust the calculation with the rate of flow in the second second, and for the third second it will be a little less, and so on.  Now, our calculations are still not that accurate, the flow would be different in the first half of the first second and the second half of the first second, so we would have to adjust for that.  But, that same reasoning would be true for the first quarter, second quarter, etc.  Integration is a process where we can sum up these tiny contributions to the water accumulated in the bucket where the intervals between the summation are made smaller and smaller so we get closer to the true result.

Combining the two ideas, if dV/dt is the rate of flow out of the leak, and V is the amount of water in the bucket after t seconds, V is the integral of dV/dt evaluated over [0,t].