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Home / How To / Integral of Sin (x): Geometric Intuition – BetterExplained

# Integral of Sin (x): Geometric Intuition – BetterExplained • 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.

## 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.  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!