how is it useful in our lives? any real-world applications?
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Integration is the opposite of differentiation, or taking a derivative, which can be thought of the instantaneous rate of change. For example, taking the derivative of speed with respect to time gives you acceleration with respect to time. So if I had a graph of your car’s speed across a stretch of time and I took the derivative of that, I would get a new graph showing your acceleration at each moment in time.
Integration reverses this process of differentiation. One way to visualize it is as the “area under the curve.” So if I had a graph of your velocity over time, if I colored in the area under the curve and could compute that area, it would tell me how much distance you covered during that time.
It’s used all the time in the real world for stuff like this. We don’t always take an integral analytically, but often approximate an integral using numerical methods in order to find out stuff like how much distance was covered.
Imagine you have a pipe. Water flows through the pipe. For the first hour, water flows through at 3 gallons/hour. For the second, it flows through at 4 gallons/hour. For the third, it flows through at 2 gallons/hour.
How much water flowed through the pipe?
Well, this is an easy problem. 3 gallons/hour times 1 hour = 3 gallons in the first hour, 4 gallons/hour times 1 hour = 4 gallons in the second hour, and 2 gallons/hour times 1 hour = 2 gallons in the third hour. That is, 3 + 4 + 2 = 9 gallons total.
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But what if the pipe works a little differently? What if instead of working at the same speed for a full hour, the flow were always changing? What if, say, the flow starts at 0 and smoothly increases to 30 gallons/hour after three hours? So after half an hour it’s flowing at 5 gallons/hr, after an hour it’s flowing at 10 gallons/hour, and so on, but it’s smoothly changing between those points. How much water does the pipe pump in those three hours?
It’s not necessarily obvious how you’d solve this problem with basic math tools. Because the rate is always changing, you can’t just use the basic (amount) = (rate) x (time) formula, because there’s never a window of time to multiply by. It only pumps at 10 gallons/hour for a single instant at exactly 1 hour, but it was pumping slightly less just before, and slightly more just afterward.
In a sense, what you’d “like” to do is be able to solve this the same way you solved the first problem, by cutting things up into intervals where the flow is constant and then just adding them up. If you could do that, the problem would be easy. And it turns out that integration is effectively that: it lets you cut the time up into *infinitely small* intervals, treat the flow as constant during those intervals, and then *add up infinitely many of those infinitely small intervals* to get a sensible answer.
The answer, as it turns out, is integral from t=0 to t=3 of the flow rate (which, since it goes from 0 to 30 in 3 hours, is 10t gallons/sec). We compute: integral(t from 0 to 3) 10t dt = 5t^(2)|(t=0 to t=3) = 5(3)^2 – 5(0)^2 = 45 gallons. Easy.
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Integration has more applications than I could list. But to name just a few:
* It allows you to compute the area or volume of objects that are more complicated than geometry would help you with. You can find something like the area under a parabola, prove the area of a circle formula from geometry, or otherwise handle shapes geometry can’t.
* It allows you to handle continuous changes in things. Position over time is the integral of velocity. The inflation since 1980 is the integral from 1980 to today of all the rates of inflation over that time. The total water pumped by our pipe is the integral of the rate at which the water is flowing.
* To find things like center of gravity in physics or center of lift in aviation.
* To compute the amount of energy required to move an object against some opposing force, like a wind or an electric field.
* To find the length of complicated curves, or even to define what we mean by “length of a curve”.
Integrals are a foundational part of almost every computational field. In fact, many formulas you’re probably familiar with (things like x(t) = -(1/2)at^2 + v0*t + x_0 for movement under constant acceleration, or even KE = 1/2 mv^2 for kinetic energy) are actually just special cases of integrals.
I’d argue integrals probably have more “bang for your buck” in terms of applications than anything else in all of math, actually. They’re often pretty easy to compute and can solve a huge number of seemingly unrelated problems.
Since you ask about the utility, did you try searching on that question already? Here are two such results:
https://www.toppr.com/bytes/calculus-in-everyday-life/
https://www.quora.com/What-are-some-real-life-applications-of-integration-and-differentiation
Places where integration occurs in settings relevant to the real world include probability (continuous probability distributions like the bell curve or normal distribution are described in terms of an integral), medical imaging technology (the math behind CT scans uses integrals such as the Radon transform), and Fourier transforms (used in the analysis of signals of all kinds).
Integration is commonly described as “computing the area under a curve”.
It is often used when you are dealing with something that goes on and on continuously (not necessarily a constant!) and you want to know when you can stop because the sum of all those somethings so far has reached a certain value (“enough” for whatever it is you’re doing).
Let’s say you are collecting rainwater. You have a large container and you’re willing to wait, but you want to know how many days you need to get a certain amount of water, say 50 liters. You know that, wherever you are, it rains heavily twice a week in the first three months, then it stops raining for six months, then it rains lightly 3 times a week in the last three months. How long before you collect 50 liters? What about 30? What if you start on October?
becoming used more and more – Integration/Area under curve is a fairly standard tool in data science and machine learning so it’s application will have indirect results for you.
Just for a bit of added information.
As for a mundane, real-world application, I used integrals to estimate the amount of yarn needed for a knitting project. Individual stitches are loops, and you can think of them like pixels. The body of the object was egg-shaped. It took a bit of refreshing my calculus enough to set up the formula, but it worked.
Area under the curve/volume under surface.
Almost everything in real life requires calculus. Perfectly straight lines don’t exist in the universe (other than like light)
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