Hi, I'm Toby and welcome to another episode of the joys of mathematics today
I'm not going to throw any formula or equations at you. I'm just going to tell you a story
Now behind the story you might be able to find hidden away in there the essence or some of the intuition behind
Calculus, but don't worry because this is not going to be a mathematical story
This is going to be a story about a guy called Bob who lives in a little Hut in
some snowy mountains
So I want you to just sit back and relax and enjoy listening to the story
Don't stress too much about trying to turn your brain on but hopefully you might learn something along the way
So, like I said, this story is set in the mountains now, it's a very windy and stormy day in the mountains
We've got a lot of wind. A couple of things happen though, which are not ideal
The first one is that one of the trees one of these little happy trees on the mountain over here is blown over
It falls, not such a happy little tree now
the second thing that happens in the windstorm
Is that a little section of the bridge which is made of logs here and connects one hill to another
Over a river it breaks away and falls
What it's left in its place is just a couple of wires that used to hold the logs together
The logs that had broken off don't fall straight down into the river
But they are blowing sideways a little bit to this grassy meadow down here
Okay, so all of this goings on sort of concerns Bob who lives in his little Hut and he comes out to check that
Everything is okay
Let's draw Bob here
Now this story is about the lengths that Bob goes through to do the right thing
now the reason he's so concerned that he comes out of his Hut is he knows that down here in the meadow where the
Wood has fallen, in this Meadow are a couple of deer who are very dear to him
they are a couple of friends of his and he's concerned he wants to make sure that
Everything is alright and that no one's been hurt down in this meadow
Now to be able to get down there and check he's going to have to cross this bridge of logs now
Obviously the problem with that is that a section of the bridge is missing
Bob's gonna have to come up with a plan
He does have one idea in mind and that is that looking at one of these fallen trees beside him on the hill
He thinks he might be able to use that tree lay it down along the wire that's left in place on the bridge
And create a little ramp for himself to walk down
Now there's only one problem that Bob really needs to consider about this plan
And that is that you know
We've got a curved bridge and if we lay a flat tree down on this wire
that part of the bridge there is going to be straight and
He doesn't want it to end up being so steep that he just slides down it and isn't able to climb back up to get
Home, you know bob has also some very old shoes that are completely worn away on the bottom
So he doesn't have much friction coming from his shoes
one thing that Bob knows from past
Experience is what kind of slopes and ramps he is able to walk up with these shoes of his now
Obviously walking along flat ground is going to be easy
Walking straight up and down a wall is going to be impossible. But what Bob does know is that exactly in between
vertical and horizontal
if I was to draw a line exactly in between them there that
Is the limit of how steep a ramp Bob can walk up with his shoes without slipping
okay, so if we know that this little flat tree lying down here on the ramp is going to be as
Steep as this or less it'll be fine. There'll be no problem
So Bob needs to check and the way that he's going to measure how steep the little tree ramp would be once
It's on the bridge is that he picks up a pebble. Now I've got one here by the camera
And he holds the pebble to his eye and he looks at the hole in the bridge and he just uses the pebble to measure
As sort of a measuring stick how many pebbles he think could fit in the area
vertically
between
the missing logs so if we had a little triangle
This is one log, this is the other log
These are the missing logs that would have fit in along here. And this is the wire that's just in their place holding them together
He measures how many pebbles he thinks with his eye could fit down here and how many could fit across here
Now what Bob measures just by using his little pebbles is that he thinks he could fit three pebbles in
This height and three pebbles in this width. What it means for us Is that our slope is
Exactly as tall as it is wide
it also means that it's exactly the same as this line here because this line here is
As tall as it is wide
So if everything is fine, and our pebbles were a good little measuring stick
Then a little straight ramp here should be just steep enough for Bob to be able to handle with his shoes
So let's go ahead with it. Bob's pretty strong and he carries the tree down to the slope and puts it in here
So now a section of the bridge is our little tree but it's pretty easy to cross over
so if you're here to learn a thing or two about
mathematics then what we just did is
differentiation we had a curved line
Which is our entire bridge and we found the slope at a point on our curved line
Just by zooming in on an area of interest and treating that area as a straight line. In fact in this case
We had a tree in there
So this section of our curve was treated as a straight line and we found the slope at that point
So let's keep going Bob walks along the bridge and comes down into the Meadow
What are in the meadow are a couple of deer. Let's try and draw
maybe one of those on
The head might be a bit hard to see, some little legs
four
I don't know If deer have a tail?
So
Bob has gone down into the meadow
And the first thing he notices is that all the deer are alive
The piece of the bridge hasn't fallen directly on to any of them. So that's really good news
However, what he does start to get concerned about is the fact that quite a lot of the grass down here is now being covered
by the section of the bridge
now
it's kind of concerning because he knows there's not too much grass down here to begin with and he starts to think, you know, maybe
with this grass gone and being
Inedible to the deer because it's covered in a part of the bridge
Maybe the deer are going to run out of food
Bob knows exactly how much grass a single deer will go through in a season so he knows how much
grass is needed to be down here in this meadow and
Because the meadow is just a rectangle. He knows how much grass was there originally
what Bob really wants to know to make sure his little friends are going to be safe is he wants to know how much of
The area of the grass is now gone
He wants to subtract this amount
from the total amount of grass
And make sure it's still enough to be able to feed the deer
If it turns out there isn't enough grass to keep them alive
Bob will probably have to call for help to move the section of the bridge out of their meadow or look for some other solution
So Bob gets down here onto the meadow and he thinks okay. I've got another problem
This part of the bridge that fell into the meadow is a really irregular shape
It's not a circle and it's not a square. Otherwise, I would be able to find its area quite easily
So he thinks well, maybe I can approximate this as a circle
Or I can use my trusty little pebble to try and measure how many circles would fit inside this
Irregular shape. Now I'm going to draw it up here to make it a bit easier for us see
So this is the shape of the part of the bridge that's fallen into the meadow
Like I said, it's quite irregular because it's not just one log
It's a few section of different logs that have fallen kind of looks like the shape of Australia now that I look at it
But that's just a coincidence. So Bob thinks okay
all I know is how to find the area of something like a pebble if it's
Approximated to be perfectly round. So I've got a few of these pebbles I can lay them out
I can get them from the riverbed and lay them along the shape and if I take my pebbles and lay them around
Maybe I can lay them on top of the shape
And I can just sort of count how many
Fit into this shape
Now that's one way you could do it you could count up the how many pebbles
Approximately equal the area of our unknown shape. Here we have one two, three, four, five six, seven eight nine
pebbles
This might be an OK approximation. If you find the area of these nine pebbles, you can just say that's the area of the shape
But you know, it's not perfect. We have some overhang here
so we're saying that this area is part of the shape when it isn't we've got
Some gaps in here between the pebbles because they don't
Perfectly fit together and whilst this is an approximation. It's very far from perfect
so what Bob can do to get a better approximation of this area is
To get some smaller pebbles now if he lays some smaller pebbles on here
There's going to be less of a gap between the pebbles so less area that's not being measured and when a pebble does
Overhang, it doesn't go too far off our area. It's still not perfect. We still have a bit of overhang
We still have a bit of gap between our pebbles, but we're getting a bit better
So there we are. That's what our new measuring technique would look like
We've still got a bit of area in here. That's not being measured correctly by these pebbles
We've still got a tiny bit of overhang, but it's definitely a lot better than what we had before
Yeah
This might be good enough
People but Bob he really cares about these deer and he doesn't want to take any chances with approximations. He wants to know exactly
Will these creatures be OK, will they have enough grass, how much of their grass has been taken away from them by the bridge?
We really want to know exactly okay, so this isn't good enough for Bob
He looks around him and he sees the riverbed and he sees it's made up of sand now sand
I guess you could say is a very tiny tiny pebble
Bob knows that the smaller the size of the item he uses
To approximate this area the more accurate it's going to be so he thinks to himself, you know, it's worth it
I'm going to get the sand Bob lays
individual grains of sand and little columns
Down the shape and he has to count exactly how many grains of sand he's placed
he knows the area of each grain of sand because they're perfectly round grains of sand and
He takes his time to lay them out one by one
grain by grain
Along the shape it takes so long for him to do this that while he's laying them down
It's no longer day
It turns to night
And then it turns to day again
And then it turns to night
but Bob keeps going because he is really dedicated to these creatures and
He wants to do what he can to be as accurate as possible. A couple of days later we are all done
it's back to Sun and we've
Completely filled our shape with grains of sand if you ask Bob how many grains of sand he laid down on that shape
He might say it seemed like an infinite number
But we were just getting very close to infinity
There's certainly hundreds of thousands of little grains of sand on that shape
But the good news is that bob has an almost perfect knowledge of the area of this shape here
So he pulls out his notepad. He works out the area of the original garden
He subtracts his now known area of the shape here
He works out how much grass would have been needed to feed a couple of these deer down in the meadow
he works it all out. Luckily, he finds that even with the amount of grass that's missing
There is still going to be just enough grass to feed everyone in the meadow
So this is a really good news and Bob feels really great
With this peace of mind that people are going to have enough to eat
with that Bob climbs back up the hill walks across the bridge across our little ramp made from a tree and he
walks home for the night
He had a pretty big excursion, but at least he's happy that everyone is ok
now what we did here when we calculated the area of the shape was actually
integration or the essence of integration. What integration means is that we can find the area of any shape by
Slicing it up into infinitely small pieces, and this is a pretty useful thing to be able to do
To find the area of any shape and to find the slope of any line, even if it's curved
So I hope you learned a thing or two from Bob's little adventures in the mountains today
Thank you for watching
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And I hope you have an absolutely mathematical day