How to Integrate with an AX? The Surprising Power of Planimeters – Visually Explained!

0:06 Welcome to another Mathologer video.

0:08 Picture this.

0:09 You’re deep in the woods,

0:11 armed with nothing but your trusty battle ax and a burning sense of purpose–

0:15 to rid the land of the ancient evil that has haunted it for generations.

0:20 That 2d monster over there, flat and fearsome:) Luckily you know what to do.

0:25 You have to calculate the area of the monster with your ax.

0:28 For this you jam the ax handle down near the center

0:33 of the beast And you mark where the blade hits the ground.

0:37 And then you trace the monster with the handle:

0:41 Center to edge, once around the edge, then back to the center.

0:52 Mark the blade’s new spot.

0:58 Now.

0:58 to find the area of the monster, and to finish it off,

1:01 you just have to multiply the distance between the marks,

1:04 about 5 cm, and the distance from the handle to the blade, about 80 cm.

1:10 5 times 80, “400 cm squared!!”, you declare.

1:14 The moment the monster hears its true area revealed, it crumbles into dust.

1:19 and peace returns to the land.

1:22 You are hailed as a hero, you marry the princess,

1:26 and yes– you live happily ever after:) Just in case you are wondering.

1:32 No, this way of calculating the area of plane shapes is not a fairy tale.

1:38 This works for ANY shape.

1:40 Again: Start with the handle near the center and mark where the blade touches.

1:44 Move to the edge, trace around, and return to the center.

1:49 Put another blade mark.

1:50 Multiply blade-to-handle length by mark distance— voilà, the area!

1:54 Well, at least roughly… considering it’s me doing

1:57 the tracing with all the precision of a caffeinated squirrel,

2:01 you probably weren’t expecting an exact value anyway, right?

2:05 Now, just like an ax, the so-called hatchet planimeter up there has a blade

2:11 at one end and a tracer at the other.

2:15 Surveyors and engineers used these to quickly estimate

2:18 areas of everything from land plots to mechanical parts.

2:21 If you’re keen to give this a try yourself

2:24 but don’t have a hatchet planimeter or an ax at hand,

2:27 don’t worry— you can repurpose all sorts of things.

2:30 A pocket knife works well.

2:32 Or a bent wire with one end flattened into a blade.

2:36 Or, if you are the proud owner of a penny farthing that works too!

2:41 At the same time hatchet planimeters were in use,

2:44 closely related precision planimeters were also

2:47 used— instruments that, at least in theory,

2:50 let you measure the exact area of any plane shape.

2:55 Here’s one I picked up on eBay a while ago.

2:59 Let’s trace the circle to see how this planimeter works.

3:03 Unlike before, we don’t start in the middle— we simply trace around the shape.

3:08 Alright, here’s a bit of common terminology I’ll be

3:13 using for the different types of planimeters we’ll encounter.

3:16 The red point— that’s the tracer.

3:18 I’ll call the black segment the arm And this is the constraint.

3:23 Why “constraint”?

3:24 Well, in the case of the ax, this point can only slide in the direction

3:29 of the blade— it’s constrained to that direction.

3:33 In the case of THIS planimeter,

3:35 this point is constrained to move along a circle with center the pole.

3:40 There, moving along a circle.

3:42 This type of planimeter is called a polar planimeter.

3:46 In the case of our ax planimeter we find the area by multiplying the length

3:51 of the arm by the distance between

3:54 where the constraint begins and where it ends.

3:57 But in the polar planimeter,

3:59 the constraint starts and ends at the same spot so clearly,

4:03 that won’t work here.

4:05 Actually, in this planimeter the second distance is recorded

4:08 by a wheel that is attached at right angles to the arm.

4:12 This wheel both rolls and slips, depending on how the tracer moves.

4:17 Take a look.

4:18 Did you see?

4:19 The more the tracer moves along the direction of the arm,

4:22 the more the wheel just slips— and the slower it rotates.

4:25 Let’s take another look.

4:27 Nifty, hmm:) Let’s call the total distance travelled by the wheel the ROLL.

4:33 Then it turns out that the area of the shape

4:36 is equal to the length of the ARM times this ROLL,

4:41 EXACTLY:) Pretty amazing, isn't it?

4:44 Before we move on, let’s take another close look what’s going on here:

4:49 First, with planimeters the convention is

4:51 to trace shapes in the clockwise direction.

4:54 That’s different from maths where the counterclockwise

4:58 direction is the default tracing direction.

5:01 Now let’s observe how exactly the ROLL is recorded.

5:05 Okay let’s roll That was the wheel turning 180 degrees

5:09 in the clockwise direction or, in terms of distance travelled,

5:13 half the circumference of the wheel, counted positive.

5:17 Okay here the wheel was spinning in the opposite direction, again 180 degrees,

5:22 and since in this case the distance travelled is counted negative,

5:26 the total distance travelled so far is 0.

5:30 Roll again.

5:31 Okay, again back to positive half the circumference of the wheel travelled.

5:36 And finished.

5:36 The final tally: about plus 0.7 of the wheel’s circumference.

5:40 Multiply that by the length of the arm— and boom, there’s your circle’s area.

5:46 Cool, huh?

5:47 Now, unlike what you’d expect, in the real instrument the wheel is actually not

5:53 attached at the tracer but close to the constraint.

5:56 There, that little white smudge that’s the wheel.

5:59 The placement of this wheel is a bit mysterious, but, as we’ll see,

6:04 it does not matter where exactly the wheel is

6:07 mounted as long as it is perpendicular to the ARM.

6:10 Case in point, here is the wheel mounted halfway along arm.

6:15 Trace again.

6:16 And we get the same roll as before:) Wonderful.

6:19 Oh, before I forget— the Mathematica animation

6:22 I just showed you is my adaptation of a piece of code originally put together

6:27 by Stan Wagon for his book Mathematica in Action.

6:30 If you’re into math(s) visuals, you’ll love this book.

6:34 Link’s in the description.

6:36 Alright it’s definitely quite miraculous that you can measure

6:39 the area of a shape by tracing its edge.

6:41 Right?

6:41 Think about it, for starters there are lots of shapes

6:45 that have the same perimeter and yet totally different areas.

6:50 Like those rectangles over there.

6:53 Hmm, same perimeter,

6:56 different areas— yet the planimeter still gets it right:) So what’s going on?

7:00 How does tracing the edge reveal what’s inside?

7:04 Well, my mission today is to explain this miracle.

7:08 Beautiful, beautiful stuff, promise:) But before we hit the maths,

7:12 let’s take a quick stroll through history.

7:23 Planimeters were heavily used from the mid-19th

7:25 to late 20th century by engineers,

7:28 surveyors, and scientists until digital tools took over in the 1990s.

7:32 The by far most popular type of planimeter were these polar planimeters.

7:39 This is a linear planimeter in which the constraint moves

7:45 along the blue straight line instead of along a circle.

7:49 This is a specialised planimeter for performing integration.

7:52 That’s another really fancy one, a so-called moment planimeter.

7:57 It’s got three separate wheels attached at specific angles,

8:01 and, apart from measuring area,

8:03 it also allows to calculate things like the center of mass of a shape.

8:08 These are digitally enhanced polar and linear

8:11 planimeters produced by the Haff company in Germany,

8:14 the same people who also built my purely analog polar planimeter.

8:18 According to the company’s website,

8:20 they still sell these digitally enhanced planimeters.

8:24 They cost around 1000 euros each.

8:27 In theory, it’s even possible to play this planimeter game on a sphere.

8:31 However, I am not sure whether any physical sphere planimeters

8:35 have ever been built:) Here’s a truly amazing book from 1951,

8:40 all about mechanical maths machines.

8:42 About a third of it is devoted

8:45 to the countless types of planimeters made over the years.

8:50 Makes for absolutely fascinating reading:) A modern day planimeter is

8:56 an app that allows you to calculate distances and areas,

8:59 for example on a map on your phone, by placing a couple of markers.

9:04 Definitely also worth checking out.

9:11 As usual here on Mathologer, the aim is to present the essence of whatever

9:15 maths we’re talking about in the most intuitive way possible.

9:19 And of course, I’ll also do that today with planimeter maths.

9:23 But just to demonstrate what’s often considered

9:25 a “great explanation”— and to highlight how much

9:28 further one can go in terms of accessibility and clarity with a bit of effort—

9:33 let me start by showing you how the planimeter is commonly explained using some

9:38 fancy calculus weapon called Green’s theorem:) Don’t

9:41 worry if you don’t understand everything I’ll say.

9:44 Just try to glimpse some bits of insight here and there,

9:47 enjoy the mathematical fireworks (and dad jokes:) and bide

9:51 your time for the super accessible parts of today’s video.

9:54 Okay, so let me prove that the polar planimeter does what I claim it does

10:00 in the special case of a polar planimeter

10:02 both of whose straight parts… are of equal length.

10:07 For that we consider this planimeter’s unit vector field.

10:11 Here the vector arrow F(x,y) is always perpendicular to the arm

10:16 when we move the tracer to the point (x,y) like this.

10:20 See how that little arrow sticking out

10:22 at the tracer is at right angles to the arm?

10:25 Same here, and here, and everywhere else.

10:28 Now, by just torturing Pythagoras a little bit,

10:31 we can figure out that F(x,y) is equal to this monstrosity.

10:38 Easy peasy.

10:39 Really:) Alright, here is the wheel.

10:42 and here is the curve that we want to trace:) remember,

10:44 we are tracing in the clockwise direction.

10:47 Alright, now let’s zoom in on an infinitesimal part of this picture.

10:53 There, as in the vector field, the orange vector is perpendicular to the arm

10:58 and the infinitesimal red vector is part of the curve.

11:02 Now as we trace along the red curve the distance

11:06 recorded by the wheel is this green distance there.

11:09 Right?

11:09 The wheel acts like a directional sensor.

11:12 It only ‘feels’ motion that’s perpendicular to the arm

11:15 but it slips when the motion is along the arm Now,

11:20 as all the real math(s) demons among you will know

11:23 the green distance is just the dot product of our two vectors.

11:27 Alright, back to the big picture:

11:30 add up all those green infinitesimal specks a la calculus,

11:35 and boom— there’s your ROLL.

11:37 Hang in there, almost done:) Enter Green’s

11:41 theorem— that slick trick I mentioned earlier.

11:43 It lets us trade that fancy integral up there

11:45 for a double integral over the region we’re enclosing.

11:49 What?

11:50 More scary integral signs, not fewer?

11:53 Why on Earth would anyone want that?

11:56 Well, just like in a mathematical fairy tale,

11:59 the curl of our Frankenstein vector field turns into a mathematical prince:

12:04 a nice, round, constant number:) And, as usual,

12:08 we can pull that constant in front of the integrals.

12:12 And what’s the value of that simplest of all double integrals up there?

12:16 Well, in fairly tale language,

12:18 that’s just the enchanted area held tight within the curve’s

12:22 embrace:) And there you have it… Area equals ARM times ROLL!

12:28 Q.E.D.

12:29 Definitely a very slick proof and a great

12:32 exercise for students who’ve just learned about Green’s theorem.

12:36 Yes, but this proof probably doesn’t quite cut it when the aim

12:41 is to explain the planimeter magic to a 10-year old:) Well,

12:45 the whole thing becomes a little bit easier when you consider a linear

12:49 planimeter instead of a polar planimeter

12:51 where the constraint scoots along the y-axis.

12:54 In particular, the vector field becomes much simpler.

12:57 Anyway, we still need something a lot

13:01 more accessible for our primary school audience:) Okay,

13:11 here is the arm of a planimeter.

13:13 First let’s attach the wheel in the middle.

13:16 Here is a nice visual way to associate what the wheel records with some area.

13:20 Positive area if the segment moves this way and negative

13:24 area if we are moving in the opposite direction.

13:27 Let me show you.

13:28 Okay let’s roll a bit, just straight down, nothing fancy.

13:32 Then, obviously, the red area swept by the arm is

13:34 just the length of the arm times this rolled distance here.

13:38 So if the curve we are tracing

13:41 includes a translation like this, then the contribution

13:44 of this translation to the final value of ARM

13:47 times ROLL is just this red swept area.

13:50 In other words, as far as this translation

13:53 is concerned Area really equals ARM times ROLL.

13:56 Clear, right?

13:56 Also, very important, as far as this translation is concerned,

14:00 it clearly does not matter where the wheel is located.

14:04 As long as it is mounted at right angles to the arm,

14:07 as we translate, the wheel will always record the same.

14:12 Okay.

14:12 Going in the opposite direction we get negative area.

14:16 Arm times rolled distance again.

14:19 Moving the arm sideways the wheel does not record

14:23 anything No area is contributed by a motion like this.

14:27 Works.

14:28 Now let’s translate at an angle.

14:30 Area of a parallelogram, that’s base times height and, of course,

14:34 the height here is exactly the rolled distance.

14:37 Works.

14:37 In fact I think it’s pretty clear that this will work

14:41 for any motion in which the arm is just translated around.

14:48 Now, for motions more complicated than pure translations,

14:53 this won’t necessarily hold.

14:55 But let’s postpone figuring out what happens in those cases.

14:59 For now, it’s enough to know that when the motion is a translation,

15:03 it contributes to the final ARM times ROLL value

15:06 as… swept… area— and that, in this special case,

15:11 the wheel’s position along the arm doesn’t matter.

15:15 Alright have a look at this rectangle.

15:19 Let’s use a linear planimeter whose constraint is moving along a line

15:24 to figure out what exactly ARM times ROLL is for this simple setup.

15:30 Now the wheel could be here or there or there or anywhere else along the arm.

15:37 For the moment, we won’t worry about where exactly.

15:40 Ready to go?

15:41 Okay, so let’s trace the rectangle.

15:43 First straight up.

15:44 That’s a translation of the arm and so this is

15:48 the area contributed to the final ARM times ROLL result.

15:53 Okay.

15:54 Now across to the right.

15:57 That’s quite a complicated motion as far as our arm is concerned.

16:01 Not sure what the wheel does there but, again,

16:04 let’s not worry about that for the moment.

16:07 Now move straight down.

16:08 Well that’s another translation and the area

16:11 contributed to the final results is this.

16:14 Okay, Now back to the start.

16:18 Can you see where I am going with this?

16:21 No?

16:21 Well, the motion of the arm just now is

16:24 the exact opposite of the one at the top.

16:28 This means that whatever the wheel recorded at the top,

16:31 was just cancelled out again here at the bottom.

16:34 How neat is that?

16:35 And that means that the final ARM times ROLL is

16:40 equal to this parallelogram area which is equal to this plus

16:48 this And since the blue area counts negative this sum

16:52 is exactly equal to the area of the rectangle.

16:56 Victory!

16:56 At least in the case of this rectangle and our linear planimeter,

17:00 we’ve just convinced ourselves that AREA equals ARM times ROLL.

17:04 AND, because only the translations contributed to the final result,

17:08 we also conclude that the exact position

17:11 of the wheel is irrelevant for this to be true.

17:14 Nice argument isn't it?

17:17 But what about more general shapes?

17:20 Well, how about this one?

17:22 A compound of four rectangles.

17:24 How can we measure the area of this compound shape with our linear planimeter?

17:31 Easy, just trace the four rectangles one after

17:34 the other and add up the resulting areas.

17:37 Hmm, yes, but how does that help?

17:40 Well, take a look at the common boundary of two of the rectangles.

17:46 There.

17:46 Now, as we trace the little rectangle on the left.

17:51 There, there, moving down,

17:53 the following area gets recorded as we trace across the common boundary.

17:59 Okay, now as we trace around the middle rectangle… there, there,

18:04 now as we move up across the common

18:07 boundary we’re moving in the opposite direction,

18:10 so the recorded area is the negative of the one

18:13 before— and so these two areas cancel each other out.

18:17 There zap!

18:18 The same sort of cancelling also happens for all other common boundaries.

18:22 And that means that we may as well remove all these common boundaries.

18:26 and just trace around the shape to get it’s area.

18:30 Neat, neat neat:) Finally, any reasonable shape… can be approximated

18:35 arbitrarily well with rectangles and so, at least intuitively,

18:38 it’s now clear why our linear planimeter always works as advertised.

18:43 Anyway, all very cool, don’t you think?

18:46 Even cooler, it turns out that the arguments here can

18:51 be easily modified to give the same result for polar planimeters.

18:55 Too nice to skip and so let me also show you

18:59 how that works:) Translate a copy of the arm over here.

19:04 Now do this.

19:05 The curvy rectangles that can be constructed like this can

19:11 replace the real rectangles in the proof just now.

19:16 Have a look.

19:17 Tracing up that’s a straight rotation.

19:19 Since we have not specified where the wheel is,

19:22 we don’t know what exactly is recorded here.

19:24 Now comes a translation that hugs the circle.

19:27 and so the area contributed is this.

19:31 Now another rotation.

19:32 Whatever is recorded here just cancels out

19:36 what was recorded during the rotation earlier.

19:39 Now another translation that hugs the circle.

19:42 Alright, so that means that the ARM times ROLL is

19:46 equal to that red area over there, plus the blue area.

19:51 And this is the same as this red plus the negative

19:55 blue area which is the area of the curvy rectangle.

19:59 Perfect, exactly what we want!

20:01 And now compound shapes… You can remove

20:04 common boundaries for the same reason as before.

20:07 And finally note that any shape can

20:09 be approximated arbitrarily well with these compound shapes.

20:13 What a magical argument, don’t you think?

20:16 But there’s still one piece of the puzzle I haven’t explained: the ax magic.

20:21 And, as it turns out, what we’ve covered so far won’t quite get us there.

20:25 For that, we need to dig a little bit deeper.

20:37 Alright, let’s trace this rectangle one more time with our linear planimeter.

20:40 Again, with those parts of the trace that are translations,

20:44 the area contribution recorded by the wheel is just the swept area.

20:49 Well.

20:49 Maybe this sweeping area business is also worth considering

20:53 for the more complicated parts of the trace, like this one.

20:57 During this movement the arm does not only translate but it also rotates.

21:02 Let’s take a closer look how area is swept when the arm rotates.

21:06 There.

21:07 So the swept area of the arm is partly positive and partly negative.

21:12 Back to tracing our rectangle.

21:14 On close inspection, during this complicated movement we also have

21:19 a mix of positive and negative areas being swept.

21:22 There, that’s all positive area.

21:23 And this is negative area.

21:25 There is an overlap where positive and negative cancel

21:28 out and so the swept area is really this.

21:32 Tracing down, we get more no-brainer swept area as before.

21:36 And then we get the same swept area as at the top, just with signs reversed.

21:43 Hmm, things cancel out again at the top

21:45 and at the bottom but then we don’t really know how

21:49 in general these complicated swept bits relate to what the wheel

21:52 records when it’s fixed at different positions along the arm.

21:56 And so, is considering the total swept area actually useful?

21:59 Not clear at this point, but let’s continue undeterred:) Take a look at this.

22:06 So we trace both curves simultaneously in the clockwise direction.

22:11 Our red tracer travels around the red curve

22:14 and our blue constraint travels around the blue curve.

22:17 Now here is what happens in terms of swept area.

22:23 Interesting, so the regions enclosed by our two curves are swept once each,

22:29 and some bits outside are swept twice.

22:32 Let’s take another close look what happens

22:35 here in terms of positive and negative area.

22:38 There that swept area is positive.

22:47 Okay so the area enclosed by the red curve counts positive.

22:51 And the area enclosed by the blue curve counts negative.

22:55 And the remaining swept area is swept once

22:57 in the positive direction and once in the negative direction,

23:01 leading to overall cancellation of the swept area there.

23:05 This means that the total signed swept area is just

23:08 the sum of the red positive area and the blue negative area.

23:12 And of course, here T and C stand for tracer and constraint.

23:17 Anyway, very cool, hmm?

23:18 We are definitely on to something.

23:21 What’s even cooler is that this stays true in general.

23:25 As long as the two curves are side-by-side and the two ends of the segment move

23:30 once around these curves in the clockwise direction

23:32 and return to their original positions, this will work.

23:36 This will work even if the curves and the movement

23:40 of the segment are a lot crazier than those up there.

23:44 For example, like this.

23:46 After we account for all the overlaps of positive and negative swept area,

23:51 we get the same picture and identity as before.

23:54 Even better, it turns out that the swept area is always ARM

23:59 times ROLL and that the position of the wheel does not matter.

24:03 Here is a quick way of seeing why this should be true,

24:08 why the swept area SHOULD be ARM times ROLL.

24:11 Here are a couple of snapshots of the arm at the start of its movement.

24:16 What’s the swept area up to this point?

24:19 Let’s take a close look.

24:21 We can approximate the motion of the arm

24:23 by subsequent tiny translations and rotations like this.

24:27 First translate.

24:28 Then rotate Translate.

24:30 Rotate.

24:31 Translate.

24:32 Rotate.

24:32 And so on.

24:34 This means that the total swept area is equal

24:39 to the total translated area plus the total rotated area.

24:46 Now take a look at this.

24:49 What’s happening here?

24:51 Well, as the sweeping on the left progresses,

24:56 the animation on the right shows the total accumulated rotated area.

25:01 At first, the arm rotates clockwise, adding positive red area.

25:05 But when the direction of rotation reverses,

25:08 blue area appears and cancels out the red.

25:10 And so it continues adding and subtracting.

25:13 Also, in the end, the arm returns

25:16 to its original position with a zero net rotation,

25:19 and so the total rotated area is also zero.

25:24 Take another look.

25:27 Again, the total rotated area is zero.

25:33 HaHa.

25:34 Victory again.

25:36 Very similar to our previous victories,

25:38 where all that remains after the complicated bits cancel out,

25:42 is the translated total area, which is really just ARM times ROLL,

25:45 no matter where the wheel is fitted.

25:48 Anyway, with a lot hand waiving, we’ve convinced ourselves that, in general,

25:53 ARM times ROLL is equal to the swept area which,

25:57 in turn, is equal to red area plus blue area,

26:00 no matter where the wheel is fixed.

26:02 Faantastic!

26:03 But now, with this formula in hand, we can explain everything,

26:08 even how measuring area with an ax works.

26:11 Ready for the finale?

26:12 Okay.

26:13 First, let’s warm up with the linear planimeter again.

26:16 In the case of a linear planimeter,

26:19 the blue region degenerates to a line which, of course, is of zero area.

26:24 And so we find again that the area of our shape is just ARM times ROLL.

26:30 Neat, and of course exactly the same argument works for polar planimeters.

26:35 In fact, the argument shows that if we

26:37 have the constraint run along any sort of curve,

26:41 not just lines and circles, this identity will hold.

26:45 Now, finally, what about our ax?

26:48 Well, let’s see.

26:56 With the ax, what’s the blue curve that the constraint traces?

27:01 Well, here we go!

27:03 Interesting:) Our curve is not closed and so our identity does not apply,

27:08 at least not to start with.

27:10 Well, don’t worry about that for the moment.

27:13 Imagine tracing the blue curve one more time with our ax,

27:17 from the start to the end.

27:19 But this time we attach the usual wheel in an unusual spot,

27:23 right at the constraint, where the blade is touching the ground.

27:28 Since the motion of the constraint is always along the blade,

27:33 that is, perpendicular to the wheel,

27:36 throughout its movement along the blue curve, the wheel will not turn AT ALL.

27:41 That’s interesting, isn't it?

27:43 Okay, so now let’s do something ingenious:) Let’s close

27:47 the blue curve by rotating the ax around the handle.

27:51 Now our identity applies.

27:53 Also, remember that the wheel did not turn at all before,

27:57 and so that means that the ROLL in our identity is

28:00 just the length of the circular arc that we just added.

28:04 That’s the only part where the wheel turns, right?

28:07 Aha, now it’s all starting to make sense, right?

28:10 Remember, what I measured in the fairy tale part

28:13 of this video was the straight line distance between those two points.

28:17 And, of course that straight line distance is a good

28:20 approximation for the length of our shallow circular arc.

28:23 So multiplying the length of the ax by the distance between the marks

28:28 to approximate the swept area is definitely a reasonable thing to do.

28:34 Yes, but what about the right side of our identity?

28:37 We want that right side to be just that red area.

28:40 Right?

28:40 Unfortunately, there is definitely some blue bits and pieces that we

28:45 have to worry about:( And so what is that blue area?

28:50 Well, after all the cancelling out of the overlaps, what remains is this.

28:54 What a surprise!

28:55 We end up with a mix of red and blue areas that partly cancel out,

29:00 leaving a result not too far away from zero.

29:05 Aha, so that’s why the ax planimeter works:) Okay,

29:10 but where does this mix of red and blue areas come from?

29:13 Well, on close inspection it all boils

29:15 down to the blue curve folding over itself,

29:18 causing different tracing directions around the three individual pointy regions.

29:22 Have a look.

29:23 Two of the three regions are traced clockwise as usual,

29:28 resulting in negative area and one is traced in the counterclockwise direction,

29:32 resulting in positive area.

29:35 There, blue arrows, clockwise, and red arrows, counterclockwise.

29:40 There are plenty more curious details to unpack here.

29:43 But the key takeaway is this: while

29:45 linear and polar planimeters measure area exactly,

29:48 the ax planimeter just gives an approximation.

29:52 So why use it at all?

29:54 Well, for one, it’s cheap and virtually indestructible— unlike its pricier,

29:58 much more delicate cousins.

30:00 And despite giving only an approximation,

30:02 the ax planimeter can deliver surprisingly accurate results when used with care.

30:07 And that brings us to one more special feature of the ax planimeter,

30:12 the fact that we are starting the tracing

30:14 inside the red curve and not on the curve.

30:16 Well, we start inside to make sure

30:19 that we get those convenient cancelling areas, making area C close to zero.

30:23 Just to compare, let me show you what happens

30:26 if we actually start with the tracer on the curve.

30:29 So the blue curve no longer folds over itself,

30:33 so no more cancelling bits, resulting in a huge error.

30:37 Alright, so, finally,

30:38 where then is the best spot inside the curve to start the trace.

30:43 Well, a detailed analysis reveals,

30:45 that starting at the center of mass ensures that those pesky

30:49 cancellation areas really do balance out as neatly as possible.

30:53 Problem solved:) Except that pinning down the exact center of mass is

30:58 at least as hard as finding the area of the curve:) But then,

31:02 close enough is definitely good enough as far

31:05 as our ax planimeter is concerned:) Alright,

31:09 the explanation using the complete swept

31:11 area that I’ve presented in this chapter

31:14 definitely gives the most insight into what

31:16 is going on with planimeters in general.

31:19 At the same time this explanation is much

31:22 more handwaivy than the explanations presented in previous chapters.

31:25 If you are interested in the gory details plus some amazing 3d counterparts,

31:29 check out the links in the description.

31:32 In particular, check out that textbook

31:35 by mathematical superstar Richard Courant over there.

31:38 This textbook appears to be the original source for making very

31:43 visual sense of all the planimeter magic using the total swept area.

31:54 To finish off, let me show you something cute.

31:57 Again it’s our ax in action, but this time it’s sweeping out this red ring,

32:02 as usual, in the clockwise direction.

32:05 What’s the area of this ring?

32:07 Prior to watching this video a tricky question but now… easy peasy.

32:10 Right?

32:11 As before, this formula applies.

32:13 However, just as in the case of the ax planimeter motion,

32:17 a wheel fixed at the blade end would not turn at all

32:20 throughout the tracing and so the total translated area must be 0.

32:25 Nifty!

32:26 And here is the total rotated area.

32:29 And so the area of the ring is equal to the area of that circle.

32:35 Quite a pleasant surprise, don’t you think?

32:37 And, of course, the same is true for any ring like that.

32:39 In particular, it’s true for all circular rings.

32:41 But that also means that all circular rings

32:44 with the same length inner cord have the same area.

32:47 All of these have the same area.

32:50 All super nice— and in fact,

32:53 this is the starting point for a beautiful theory of doing

32:58 calculus without actually doing calculus:)

33:00 beautifully explained in this remarkable book.

33:03 On my to-do list for another video:) Anyway,

33:07 that’s it for today:) Except for one final challenge for you.

33:13 Take a look at this— what’s the red area under the blue curve?

33:18 Share your answers in the comments below!

33:21 And so, until next time— take care:)