Explain Integration to me, please

[Alienspecimen]

Hi all,

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.

 

[jedishrfu]

This is where dimensional analysis can come into play to help you interpret the area under the curve:

- if the x is in meters ($L$) and the y is in meters ($L$), then the area is meters^2 ( $L^2$)
- if the x is in meters (length) and the y is in newtons ($ML / T^{-2}$), then the area is joules ( $ML^2 / T^{-2}$ ), which is work

In your cases, try using dimensional analysis to see if it makes sense. Here's a video on the topic

 

[berkeman]

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

Without integration, how would your power company know how much to charge you each month...?

https://en.wikipedia.org/wiki/Electricity_meter

 

[Mark44]

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.

An integral can be thought of as the continuous counterpart to a summation, which is an operation performed on discrete quantities. It can also be thought of as an inverse operation to differentiation. Some physical properties are the derivatives of other physical properties. For example, current i is the time rate of change of charge q. I.e. $i(t) = \frac{dq}{dt}$. Also force F is the time rate of change of momentum p -- $F = \frac{dp}{dt}$. You can use integration to write the inverse relationships here.

Calculating an area is only one of many possible applications of integrals. Other applications include the probabilities of continuous distributions and calculations of work done, and many more. Most calculus textbooks provide lots of examples of "real world" applications of integration.

Without integration, how would your power company know how much to charge you each month...?

My electric meter works in a way that is similar to how my water meter works. As the water flows past the meter, the water causes vanes to rotate that are connected to gears that count the number of rotations. The more water used, the faster the vanes turn. I'm reasonably sure the electric meter works about the same way -- the more current flows past the equivalent of vanes in a water meter, the faster the numbers increase.
The meter reader doesn't need to use calculus at all to report electricity usage.

 

[berkeman]

My electric meter works in a way that is similar to how my water meter works. As the water flows past the meter, the water causes vanes to rotate that are connected to gears that count the number of rotations. The more water used, the faster the vanes turn. I'm reasonably sure the electric meter works about the same way -- the more current flows past the equivalent of vanes in a water meter, the faster the numbers increase.
The meter reader doesn't need to use calculus at all to report electricity usage.

Yeah, the older ones like I showed in the photo work like that, but it's still a form of integration. And the more modern meters that I've worked on digitize the voltage and current to calculate power as a function of time, and integrate that to get kWhr energy consumption.

 

[Muu9]

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?

The total "volume-seconds" occupied by the water over the interval. If you divide by the length of the interval in seconds, you can calculate the average volume of the tank.

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?

The displacement. Specifically, the indefinite integral would be a function that takes as an input some time and whose output represents how far the actuator extended from t=0 to the time in question. The definite integral over some interval of time would be how far the actuator extended during that time interval