Take a look at the demo of the falling apple from the differentiation page again. The rate of change of the position is velocity, and the rate of change of velocity is acceleration. We can think about this the other way around. The position of an object at a certain moment is a result of the effect of velocity accumulated over time, and the velocity at a certain moment is a result of the effect of acceleration accumulated over time.



For example, if a car has been moving at 100 km/h toward east for 2 hours, it must be at 200 km east from its starting point. This is the basic idea of integration. An integral represents the accumulation of the rate of change over a certain span. Differentiation relates to 'difference', capturing the rate of change at each moment, while Integration ties to 'sum', accumulating these changes over time. In fact, the origin of the symbol of integral $\int$ is the elongated "S," which stands for "summa" in Latin, meaning "sum" or "total.”


<aside> 💡 Integration doesn't always have to be about position and time, but we'll stick with this example for now as it is the most intuitive. 積分は必ずしも時間や位置に対する問題でなくて良いのですが、今はこの例を最も直感的な例として用いることにします。


Area and Integration


Integration can also be thought of as the area under the graph. Let's consider the relationship between velocity and position as an example. If the velocity is constant, the change in position is simply the product of velocity and time. It is clear that it corresponds to the area of the shaded part of the graph below.



If we plot the graph of the area, it will look like this. This is the graph of the integral of velocity, which represents the change in position over time. When the velocity is constant, the position graph becomes a straight line.



It may not be that obvious when the graph is not straight. But if you imagine that the shape consists of many narrow rectangle strips, you can approximate its area. Integration is the accumulation of change over time. You can think of each strip as an approximation of the effect of velocity over a short amount of time. By summing up the areas of all the strips, you can find the total amount of change.