15.1Changing Circumference To Area

Our first example of “step-by-step” thinking was gluing a sequence of rings to make a circle:

When we started, we needed a lot of visualization.We had to unroll the rings, line them up, realize they made a triangle, then use\( \frac{1}{2} \textit{base} \cdot \textit{height} \)to get the area.Visual, tedious… and necessary.We need to feel what’s happening before working with raw equations.

Here’s the symbolic approach:

Let’s walk through it.The notion of a “ring-by-ring timelapse” sharpens into “integrate the rings, from nothing to the full radius” and ultimately:

Each ring has height\( 2 \pi r \)and width\( dr \)，我们想要积累这个区域来形成我们的圆盘。

How can we solve this equation?By working backwards.We can move the\( 2 \pi \)积分外的部分(还记得缩放性质吗?)and focus on the integral of\( r \):

What pattern makes steps of size\( r \)?Well, we know that\( r^2 \)creates steps of size\( 2r \), which is twice what we need.Half that should be perfect.Let’s try it out:

Yep,\( \frac{1}{2}r^2 \)gives us the steps we need!Now we can plug in the solution to the integral:

This is the same result as making the ring-triangle in the first lesson, but we manipulated equations, not diagrams.Not bad!It’ll help even more once we get to 3d…

15.2Changing Area To Volume

让我们更漂亮。We can take our discs, thicken them into plates, and build a sphere:

Let’s walk slowly.We have several plates, each at a different “x-coordinate”.What’s the size of a single plate?

The plate has a thickness (\( dx \)), and its own radius.The radius of the plate is its height from the x-axis, which we can call\( y \).

It’s a little confusing at first:\( r \)is the radius of the entire sphere, but\( y \)is the (usually smaller) radius of an individual plate under examination.In fact, only the center plate (\( x=0 \)) will have its radius the same as the entire sphere.The “end plates” don’t have a height at all.

And by thePythagorean theorem, we have a connection between the x-position of the plate, and its height (\( y \)):

Ok.We have size of each plate, and can integrate to find the volume, right?

Not so fast.Instead of starting on the left side, with a negative x-coordinate, moving to 0, and then up to the max, let’s just think about a sphere as two halves:

To find the total volume, get the volume of one half, and double it.这是一个常见的技巧:如果一个形状是对称的，取其中一个部分的大小，并将其放大。Often, it’s easier to work out “0 to max” than “min to max”, especially when “min” is negative.

Ok.Nowlet’s solve it:

Whoa!Quite an equation, there.It seems like a lot, but we’ll work through it:

First off, three variables (\( r \),\( y \),\( x \)) is too many to have flying around in a single equation.We’ll write the height of each plate (\( y \)), in terms of the others:

The square root looks intimidating at first, but it’s being plugged into\( y^2 \)and the exponent will cancel it out.After plugging in y and moving\( \pi \)outside the integral, we have the much nicer:

The parentheses are often dropped because it’s understood that\( dx \)is multiplied by the entire size of the step.We know the step is\( (r^2 - x^2) dx \)and not\( r^2 - (x^2 dx) \).

Let’s talk about\( r \)and\( x \)一分钟。\( r \)is the radius of the entiresphere, such as “15 inches”.You can imagine asking “I want the volume of a sphere with a radius of 15 inches”.Fine.

To figure this out, we’ll create plates at each x-coordinate, from\( x=0 \)up to\( x=15 \)(双)。\( x \)is the bookkeeping entry that remembers which plate we’re on.We could work out the volume from\( x=0 \)to\( x=7.5 \), let’s say, and we’d build a partial sphere (maybe useful, maybe not).But we want the whole shebang, so we let\( x \)go from 0 to the full\( r \).

Time to solve this bad boy.What equation has steps like\( r^2 - x^2 \)?

First, let’s use the addition rule: steps like\( a - b \)are made from two patterns (one making\( a \), the other making\( b \)).

Let’s look at the first pattern, the steps of size\( r^2 \).We’re moving along the x-axis, and\( r \)is a number that never changes: it’s 15 inches, the size of our sphere.This max radius never depends on\( x \)，电流板的位置。

When a scaling factor doesn’t change during the integral (\( r \),\( \pi \), etc.), it can be moved outside and scaled up at the end.So we get:

In other words,\( r^2 \cdot x \)is a linear trajectory that contributes a constant\( r^2 \)在每一个步骤。

酷。How about the integral of\( -x^2 \)?First, we can move out the negative sign and take the integral of\( x^2 \):

我们以前见过。Since\( x^3 \)has steps of\( 3x^2 \), taking 1/3 of that amount (\( \frac{x^3}{3} \)) should be just right.And we can check that our integral is correct:

它工作了!随着时间的推移，你会学会相信你逆向工程的积分，但在开始时，最好检查一下导数。With the integrals solved, we plug them in:

What’s left?Well, our formula still has\( x \)inside, which measures the volume from 0 to some final value of\( x \).In this case, we want the full radius, so we set\( x=r \):

Tada!You’ve found the volume of a sphere (or another portion of a sphere, if you use a different range for\( x \)).

Think that was hard work?You have no idea.That one-line computation took Archimedes, one of the greatest geniuses of all time, tremendous effort tofigure out.他必须想象一些球体，一个圆柱体，一些圆锥体和一个支点，想象它们平衡，这么说吧，当他找到公式时，他把它写在了他的坟墓上。Your current intuition would have saved him incredible effort (see thisvideo).

15.3Changing Volume To Surface Area

Now that we have volume, finding surface area is much easier.We can take a thin “peel” of the sphere with a shell-by-shell X-Ray:

I imagine the entire shell as “powder” on the surface of the existing sphere.有多少粉末?It’s\( dV \), the change in volume.Ok, what is theareathe powder covers?

Hrm.Think of a similar question: how much area will a bag of mulch cover?Get the volume, divide by the desired thickness, and you have the area covered.If I give you 300 cubic inches of dirt, and spread it in a pile 2 inches thick, the pile will cover 150 square inches.After all, if\( \textit{Area} \cdot \textit{Thickness} = \textit{Volume} \)then\( \textit{Area} = \frac{Volume}{Thickness} \).

In our case,\( dV \)is the volume of the shell, and\( dr \)是它的厚度。We can spread\( dV \)along the thickness we’re considering (\( dr \)) and see how much area we added:\( \frac{dV}{dr} \), the derivative.

This is where the right notation comes in handy.We can think of the derivative as an abstract, instantaneous rate of change (\( V' \)), or as a specific ratio (\( \frac{dV}{dr} \)).In this case, we want to consider the individual elements, and how they interact (volume of shell / thickness of shell).

So, given the relation,

we figure out:

Wow, that was fast!The order of our morph (Circumference\( \rightarrow \)Area\( \rightarrow \)Volume\( \rightarrow \)Surface area) made the last step simple.We could try to spin a circumference into surface area directly, but it’smore complex.

As we cranked through this formula, we “dropped the exponent” on\( r^3 \)to get\( 3 r^2 \).Remember the total change comes from 3 perspectives that contribute an equal share:\( \frac{d}{dr} r^3 = r^2 + r^2 + r^2 = 3r^2 \).

15.42000 Years Of Math In A Day

我们历经了2000年的思考才发现了这些步骤，而这些都是出自最伟大的天才之手。Calculus is such a broad and breathtaking viewpoint that it’s difficult to imagine where itdoesn’tapply.It’s just about using X-Ray and Time-Lapse vision:

• Break things down.In your current situation, what’s the next thing that will happen?And after that?Is there a pattern here?(Getting bigger, smaller, staying the same.)Is that knowledge useful to you?
• Find the source.You’re seeing a bunch of changes – what caused them?If you know the source, can you predict the end-result of all the changes?Is that prediction helpful?

We’re used to analyzing equations, but I hope it doesn’t stop there.数字可以描述情绪、辣度和客户满意度;按部就班的思考可以描述作战计划和心理治疗。方程和几何只是分析的好起点。数学不是方程，音乐也不是乐谱——它们指向了符号内部的思想。

While there are more details for other derivatives, integration techniques, and how infinity works, you don’t need them to start thinking with Calculus.你今天的发现会让阿基米德热泪盈眶，这对我来说是一个足够好的开始。

Happy math.