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Integral of Sin (x): Geometric Intuition – BetterExplained

You think of your own business when some punk asks what the integral of $ sin (x) $ means. Your options:

  • Pretend to sleep (except not in the library again)
  • Preserved answer: "As with all functions, the integral of the sinus area is below its curve."
  • Geometric intuition: "The integral of the sine is the horizontal distance along a circular path."

Option 1 is tempting, but let's take a look at the others.

Why "Area Under the Curve" is unsatisfactory [19659007] Describing an integral as "area under the curve" is like describing a book as a list of words. Technically correct, but miss the message and I suspect you haven't done the assigned read.

If you are not trapped in LegoLand, integrals mean anything except rectangles.

Decoding the integral

My spreadsheet space did not have an intuition for all mechanics.

When we see:

$ int sin (x) dx $

We can get some insights:

  • The integral is just a nice multiplication. Multiplication collects unchanged numbers (3 + 3 + 3 + 3). Integrals add numbers that can be changed based on a pattern (1 + 2 + 3 + 4). But if we blink our eyes and pretend that objects are identical, we have a multiplication.

  • $ sin (x) $ just a percentage. Yes, it's also a nice curve with nice features. But at any time (like 45 degrees) it is a single percent from -100% to + 100%. Just ordinary numbers.

  • $ dx $ is a small, infinite part of the path we take. 0 to $ x $ is all the way, so $ dx $ is (intuitively) a nanometer.

Ok. With the three intuitions, our rough (coarse!) Conversion to plain English is:

The integral of its (x) multiplies our intended path length (from 0 to x) by a percentage

We refer to for to travel an easy route from 0 to x, but we end up with a smaller percentage instead. (Why? Since $ sin (x) $ is usually less than 100%). So we can expect something like 0.75x.

If $ sin (x) $ actually had a fixed value of 0.75, our integral would be:

$ int text {fixedsin} (x) = int 0.75 dx = 0.75 int dx = 0.75x $

But the real $ sin (x) $, the squirrel villain, changes as we go. Let's see what fraction of our road we really get.

Visualize the change in sin (x)

Now let's visualize $ sin (x) $ and its changes:

Here is the decoder key:

  • $ x $ is our current angle in radians. On the unit circle (radius = 1), the angle is the distance along the circumference.

  • $ dx $ is a small change in our angle, which becomes the same change along the circumference (movement 0.01 units in our angle moves 0.01 along the circumference.)

  • On our small scale, a circle is a polygon with many sides, so we move along a line segment of length $ dx $. This puts us in a new position.

With me? With trigonometry, we can find the exact change in height / width as we glide along the circle with $ dx $.

With similar triangles, our change is only our original triangle, rotated and scaled.

  • Original triangle (hypotenuse) = 1): height = $ sin (x) $, width = $ cos (x) $
  • Change triangle (hypotenuse = dx): height = $ sin (x) dx $, width = $ cos (x) dx $

Remember that sine and cosine are functions that return percentages. (A number like 0.75 does not have its orientation. It turns up and does things 75% of their size in whatever direction they are facing.)

So given how we have drawn our triangle for change, $ sin (x) dx $ is our horizontal change. Our common English intuition is:

The integration of its (x) adds to the horizontal change along our path

Visualize The Integral Intuition

Ok. Let's graph this bad boy to see what happens. With our "little horizontal change = $ sin (x) dx $ = little horizontal change" insight we have:

When we circle around we have a bunch of line segment $ dx $ (in red). When the sine is small (around x = 0) we hardly get any horizontal movement. As the sine becomes larger (the top of the circle), we move up to 100% horizontally.

In the end, the different segments $ sin (x) dx $ move us horizontally from one side of the circle to the other. [19659011] A more technical description:

$ int_0 ^ x sin (x) dx = text {horizontal distance traveled on the arc from 0 to x} $

Aha! That's the point. Let's eyeball. When we move from $ x = 0 $ to $ x = pi $, we move exactly 2 units horizontally. It makes perfect sense in the chart.

The Official Calculation

Using the official calculation facts as $ int sin (x) dx = – cos (x) $ we would calculate:

$ int_0 ^ pi sin (x ) dx = – cos (x) Big | _0 ^ pi = – cos ( pi) – – cos (0) = – (- 1) – (- 1) = 1 + 1 = 2 $

Yowza. Do you see how difficult it is, these double negatives? Why was the visual intuition so much easier?

Our path along the circle ($ x = 0 $ to $ x = pi $) moves from right to left. But the x-axis goes positive from left to right. As we transform distances along our path to Standard Area ™, we must turn our shoulders:

Our excitement to put things in the official format evaporated the intuition of what happened.

Fundamental Theorem of Calculus

We are not really talking about the basic theorem of Calculus. (Is that something I did?)

Instead of posting all the small segments, just do: endpoint – starting point.

Intuition stared us in the face: $ cos (x) $ is the anti-derivative, and traces the horizontal position, so we only make a difference between horizontal positions! (With troublesome negatives to change their shoulders.)

That is the power of the basic theory of calculation. Skip the intermediate steps and subtract only endpoints.

Forward and Upward

Why did I write this? Since I could not directly calculate:

$ int_0 ^ pi sin (x) dx = 2 $

This is not an exotic function with strange parameters. It's like asking someone to calculate $ 2 ^ 3 $ without a calculator. If you claim to understand exponents, should that be possible, right?

Now we can't always visualize things. But for the most common functions we owe a visual intuition. I really can't eyeball the two ranges from 0 to $ pi $ under a sine curve.

Happy math.

Appendix: Average Efficiency

As a fun fact, the "average" efficiency for motion around the top of a circle (0 to $ pi $) is: $ frac {2} { pi} = .6366 $

So, on average, 63.66% of your orbit's length is converted to horizontal motion.

Appendix: Height controls width?

It seems strange that the height controls the width, and vice versa, right? But a circle must regulate itself.

$ e ^ x $ is the child who eats sweets, gets bigger and can therefore eat more sweets.

$ sin (x) $ is the child who eats sweets, gets sick, waits for appetite and eats more sweets.

Appendix: The area is not literal

The "area" in our integral is not the literal range, it is a percentage of our length. We visualized the multiplication as a 2d rectangle in our general integral, but it can be confusing. If you buy an item and are taxed, do you visualize the 2d area? Or just a 1d quantity that shrinks?

Range mainly indicates that a multiplication occurs, especially since $ sin (x) $ returns a modest scalar number. Don't let Team Integrals Are Area win every fight!

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