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I understand what the integral does - it calculates the area under a curve and can easily see how it could be used to calculate an area of land. What I do not understand is really the physical meaning when it comes to the real world. Here are some examples:

1. A set of data representing the change of the voltage of an electrical signal with time. From the data we find that the change with time follows a pattern that could be described by a mathematical function. I do understand that every point on the curve once that function is plotted represents a measure of the voltage at that particular instance. If I were to take the integral of the function either throughout the entire interval when the voltage was recorded, or just a sub-interval, what did I just calculate/quantify?

2. I have a large container full of water. The level of the water changes with time. Eventually, just as in the example above, I am able to find a pattern and describe the change of water level with time with a function. Each point of the graph represents the level of the water in the tank at that particular time. If I took the integral of the function, what did I calculate?

3. An actuator moves in a straight line. At the starting point the actuator starts moving at certain speed V, but as it moves, its speed gradually decreases and it finally stops once fully extended. As with the previous two examples, V with respect to time could be described with a function. What does the integral of the function represent?

Can you please help me relate integration to real world problems?

Thanks in advance.