Integral of Sin(x): Geometric Intuition

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

Pretend to be asleep (except not in the engineering library again)
可以这样回答:“与任何函数一样，正弦函数的积分是其曲线下的面积。”
Geometric intuition: "The integral of sine is thehorizontal distancealong a circular path."

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

Why "Area Under the Curve" is Unsatisfying

Describing an integral as "area under the curve" is like describing a book as a list of words. Technically correct, but misses the message and I suspect you haven't done the assigned reading.

除非你被困在乐高乐园里，否则积分的意义不仅仅是矩形。

Decoding the Integral

My calculus conundrum was not having an intuition for all the mechanics.

When we see:

$\int \sin(x) dx$

We can call on a few insights:

The integral is justfancy multiplication. Multiplication accumulates numbers that don't change (3 + 3 + 3 + 3). Integrals add up numbers thatmightchange, based on a pattern (1 + 2 + 3 + 4). But if we squint our eyes and pretend items are identical we have a multiplication.
$\sin(x)$ just apercentage. Yes, it's also fancy curve with nice properties. But at any point (like 45 degrees), it's a singlepercentagefrom -100% to +100%. Just regular numbers.
$dx$ is a tiny,infinitesimal partof the path we're taking. 0 to $x$ is the full path, so $dx$ is (intuitively) a nanometer wide.

Ok. With those 3 intuitions, our rough (rough!) conversion to Plain English is:

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

Weintend从0到x的简单路径，但是我们得到的百分比更小。(为什么?因为$\sin(x)$通常小于100%)。所以我们期望得到0.75x。

In fact, if $\sin(x)$ did have a fixed value of 0.75, our integral would be:

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

但是真正的$\sin(x)$，这个无赖，随着我们的进行而变化。Let's see what fraction of our path we really get.

Visualize The Change in Sin(x)

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

Here's the decoder key:

$x$ is our current angle inradians. On the unit circle (radius=1), the angle is the distance along the circumference.
$dx$是角度的微小变化，也就是圆周上的变化(角度移动0.01个单位，圆周上移动0.01个单位)
At our tiny scale, a circle is aa polygon with many sides, so we're moving along aline segmentdx美元的长度。这将我们置于一个新的位置。

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

By similar triangles, our change just just 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 = $\cos(x) dx$

Now, remember that sine and cosine are functions that return percentages. (A number like 0.75 doesn't have its orientation. It shows up and makes things 75% of their size in whatever direction they're facing.)

So, given how we've drawn our Triangle of Change, $\sin(x) dx$ is our horizontal change. Our plain-English intuition is:

The integral of sin(x) adds up the horizontal change along our path

Visualize The Integral Intuition

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

As we circle around, we have a bunch of $dx$ line segments (in red). When sine is small (around x=0) we barely get any horizontal motion. As sine gets larger (top of circle), we are moving up to 100% horizontally.

Ultimately, the various $\sin(x) dx$ segments move us horizontally from one side of the circle to the other.

A more technical description:

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

Aha! That's the meaning. Let's eyeball it. When moving from $x=0$ to $x=\pi$ we move exactly 2 units horizontally. It makes complete sense in the diagram.

The Official Calculation

Using the Official Calculus Fact that $\int \sin(x) dx = -\cos(x)$ we would calculate:

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

Yowza. See how awkward it is, those double negations? Why was the visual intuition so much simpler?

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. When convert distance along our path into Standard Area™, we have to flip our axes:

Our excitement to put things in the official format stamped out the intuition of what was happening.

Fundamental Theorem of Calculus

We don't really talk about theFundamental Theorem of Calculusanymore. (Is it something I did?)

Instead of adding up all the tiny segments, just do: end point - start point.

The intuition was staring us in the face: $\cos(x)$ is the anti-derivative, and tracks the horizontal position, so we're just taking a difference between horizontal positions! (With awkward negatives to swap the axes.)

That'sthe power of the Fundamental Theorem of Calculus. Skip the intermediate steps and just subtract endpoints.

Onward and Upward

我为什么要写这个?Because I couldn'tinstantly figure out:

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

This isn't an exotic function with strange parameters. It's like asking someone to figure out $2^3$ without a calculator. If you claim to understand exponents, it should be possible, right?

Now, we can't always visualize things. But for themost commonfunctions we owe ourselves a visual intuition. I certainly can't eyeball the 2 units of area from 0 to $\pi$ under a sine curve.

快乐数学。

Appendix: Average Efficiency

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

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

Appendix: Height controls width?

高度控制宽度似乎很奇怪，反之亦然，对吧?

If height controlled height, we'd have runawayexponential growth. But a circle needs to regulate itself.

$e^x$表示孩子吃了糖果，长大了，因此可以吃更多的糖果。

$\sin(x) $ is the kid who eats candy, gets sick, waits for an appetite, and eats more candy.

Appendix: Area isn't literal

The "area" in our integral isn't literal area, it's a percentage of our length. We visualized the multiplication as a 2d rectangle in our generic integral, but it can be confusing. If you earn money and are taxed, do you visualize 2d area (income * (1 - tax))? Or just a quantity helplessly shrinking?

Area primarily indicates a multiplication happened. Don't let team Integrals Are Literal Area win every battle!