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I understand Integration and Differentiation and see a lot of Physics / Electrical Theory using them.

Take for example a sine wave. So area for me means the space any object would occupy. So what's usage it comes to find the area of a sine curve?

There are so a many formulas that calculates the area by Integration - but why calculation is required - I mean what information we can get (isn't it just space occupied) or rather what data we can find by calculating area of a curve via Integration?

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  • $\begingroup$ You know speed of pendulum by $v=A\omega\sin(\omega t+\phi)$. Integrate within proper limits wrt time and get your distance as a function of time! $\endgroup$ – evil999man May 2 '14 at 3:48
  • $\begingroup$ Thanks but I think I having some basic understanding issues - Its area right - how come for our example its is distance as function of time. $\endgroup$ – Programmer May 2 '14 at 3:53
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    $\begingroup$ Usually it's not the fact that we are finding the area that is interesting, per se, it is the quantity that area represents given the situation. $\endgroup$ – wgrenard May 2 '14 at 3:56
  • $\begingroup$ Correct - both are same - area and quantity that area represents given the situation. Then in the end of Integral we know it - why it so required in so many theories. As we all know that Integration is used in so many many formulas - using the area calculated what information other than its area it provided? $\endgroup$ – Programmer May 2 '14 at 4:04
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    $\begingroup$ Area under the graph is only one interpretation of integration, and is not necessarily the best way to think of it. One reason why integration comes up so often is because we want to add up all the little changes in a quantity to get the total change. $\endgroup$ – littleO May 2 '14 at 4:55
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Based off of your question and the subsequent comments you have made I think I understand what you are looking for. I will give an example of what an integral can tell us, in addition to providing the area under a curve. Keep in mind, this is only one of very many examples.

We know that near the Earth's surface, an object in free fall accelerates at approximately $9.8 {m \over s^2}$. We can plot this acceleration as a function of time, and the resulting graph will look like a horizontal line. If I take the integral of this function from one point in time to another (say from $0$ to $10$ seconds), the result that I obtain is the change in velocity between those two points in time: $$ a(t) = 9.8$$ $$ \int_0^{10} 9.8dt=98 $$ What this result tells me is that if I release an object in the air and let it fall vertically for $10$ seconds it will undergo a change in velocity such that $\Delta v = 98 {m \over s}$. It also tells me the area from $t=0$ to $10$ under that horizontal line that is my acceleration graph, is $98$. As you can see, when I calculated this integral, I wasn't interested in finding the area under the acceleration graph. I was interested in finding the change in velocity. It just so happens to be that the two are the same thing; the area represents the change in velocity. This is one example of how the integral can tell you more than just the area under the graph.

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The point is that in science and in real life, you come up against situations where you need to totalize or to aggregate or to accumulate some kind of growing quantity. Say you know how fast the snow is falling at any particular time, and want to know the accumulation after five hours. Say you have pollutants flowing into a lake at varying rates during the day or week, and want to know how much crud is in the lake after a few weeks. Both these examples are quantities that are accumulated from a varying contribution, and both are measured by an integral. And there’s no area in sight.

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  • $\begingroup$ Thanks for the explicit clarification - then why Integral is said to find area under a curve. It does means that it has more to add in its definition. Please let me know the various data we can find from Integration apart from area $\endgroup$ – Programmer May 2 '14 at 4:08
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    $\begingroup$ If you’ve seen the proof of the Fundamental Theorem of Calculus, you’ve seen how the area is collected or aggregated as the vertical line sweeps from the lefthand endpoint to the righthand endpoint. For other applications of the integral, I’d much rather that you use your own imagination. Think of varying interest rates on your savings account, for instance. Once you look, integration is everywhere. $\endgroup$ – Lubin May 2 '14 at 4:34
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Suppose water is flowing into a little pond at a rate of 5 liters per second. Over the course of 10 seconds, the amount of water in the pond will increase by $5 \tfrac{\text{L}}{\text{s}} \times 10\,\text{s} = 50\,\text{L}$.

If you graph the flow rate, the graph will be a horizontal line $5 \tfrac{\text{L}}{\text{s}}$ above the horizontal axis. The region between the graph and the horizontal axis from $0\,\text{s}$ to $10\,\text{s}$ will be a rectangle with area $5 \tfrac{\text{L}}{\text{s}} \times (10 - 0)\,\text{s} = 50\,\text{L}$, which is exactly the amount of water that flowed into the pond over that $10\,\text{s}$ period. This isn't a coincidence: you should be able to convince yourself, maybe with the help of a mathematically-inclined friend, that even if the flow rate is changing over time, measuring the area under the flow rate graph over a certain time period will always tell you how much the amount of water in the pond increased over that period. {1}

If you make the convention that area below the horizontal axis counts as negative, the statement above will stay true even if the flow rate is allowed to go negative.

This relationship between the area under a flow rate graph and the total amount of stuff that has flowed is super deep. It's called the fundamental theorem of calculus, it's been discovered and rediscovered many times by many brilliant mathematicians (including Isaac Newton!), and its generalizations help hold up the foundations of much of modern math. For example, Stokes' theorem (mentioned at the bottom of that Wikipedia article) is one of the most important tools in modern geometry.

As @Lubin mentioned, there are lots of situations in which you know how fast some kind of stuff has been accumulating over time, and you want to figure out how much stuff has accumulated over a certain period. Using the connection between area and total accumulation—that is, the fundamental theorem of calculus—you can calculate the total accumulation easily!


{1} Unless the graph jumps around in such a ridiculous way that you can't even measure the area under it...

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The integral is used for far more than just computing areas of curves. However, if you do not have the intuition that comes from practicing with the idea in this form, you will have trouble applying integration in the future.

  • Integrate jerk over time, get acceleration.
  • Integrate acceleration over time, get velocity.
  • Integrate velocity over time, get position.
  • Integrate outward directed electrical flux over a surface, get enclosed charge.
  • Integrate distance from an axis times density over space, get moment of inertia around that axis.
  • Integrate blackbody radiation over frequency, get emitted power per square meter per steradian.
  • Integrate magnetic field over a loop, get the current flowing through the loop.
  • Integrate all possible future histories of a particle over the action, get its transition probability matrix (S-matrix).
  • Integrate the wavefunction of a particle times the conjugate of a measurement over the phase space, get the probability of measuring that particle in that state.

... and those are just uses in Physics. There are many more in Mathematics.

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The integral is how you get how much something changed (the function), knowing how it changes little by little (the derivative).

If you think about what science does, it's only able to measure derivatives. For example, you don't know "how fast" something is in reality (since we're moving all the time! Does "how fast" even make sense?) and you probably cannot even measure it, but you can make the change in the speed of an object that occurs by putting in $x$ force. Thus all of the laws of physics which we can measure and which are tested in labs are statements about derivatives, statements about change. What you want to know is, given how things change, what is the final result? This mapping from derivatives to antiderivates is the integral (or more generally the solution to differential equations) which is why it is so central to modern science.

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A function, differentiate it represent how fast it changes, integrate it represent how much change had accumulated.

If we know the derivative of the function (how fast it changes), by integrate it, we find function itself. If we know how much change had accumulated, by differentiate it, we can find of the function of this thing.

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