 So in this video Consider a dam that it's shaped like a trapezoid as you can see in the image provided The height of the of the dam is going to be 20 meters tall what you see right here the width of The of the dam and well, it's dependent on where you are it's 50 meters across the top and it's 30 meters across the bottom and We want to determine. What is the force on the dam due to the hydrostatic pressure? If the water level is four meters from the top of the dam so the water Comes up here. It doesn't quite reach the top. There is this four meter gap right here And so we're interested in how much force is being exerted against the dam Based upon this the hydrostatic pressure here. So let's let's try to unravel this terminology a little bit So when you see something when you see this word hydrostatic, what does that mean? Hydro we probably are familiar with hydro comes From the greek, I believe Which means water right? and so although And hydrostatic problems the fluid in play does not necessarily have to be water And it's a very abundant fluid on earth if you've ever noticed and so hence these cyber problems are named after water Static on the other hand would be essentially the antonym to dynamic here static would suggest that things are not Moving in play right we don't have like a river flowing into the dam We just have a lake or a reservoir that's sitting there. There's no there's no flow in play here It's just sitting here. So why is there any force at all if things aren't moving? Well Uh scientists like to model uh fluid fluid dynamics and in this case fluid statics Based upon the molecules that these these water is made out of you kind of think it was like a bag of marbles, right? So imagine we have a bunch of marbles that are stacked on top of each other Making some type of like pyramid of marbles of some kind. So if we're supporting the marbles like in our hand Uh, you'll you'll have like this ball the sphere of marbles, but if we were to place on the table What typically happens? Well, what happens is we we see that the marbles will start moving to outward, right? Our tower starts collapsing and it spreads out. Why is it doing that? Um, why what's causing these marbles to spread to the sides? Well, the basic idea is that the marbles on top while they're being pulled down by gravity And when we place them on the table the table itself is exerting a normal force back to them And so they can't just sink through the through the table But because of the these these round marbles your the marbles on top are kind of squishing the ones on bottom And so that pushes them to to go outward, right? And so they spread out like so and so this we've seen this if we've ever played with marbles or small balls before Uh, but it turns out on the the molecular level, uh, are these molecules are working At a very similar in a very similar way than this as the water molecules above Are coming down they're being pulled by gravity. They're actually pushing outward and so in In fluid dynamics when we have a molecule, right because of its depth there's some pressure going on here So it's going to be crushing things below it But we also see that that pressure is going to be equal in all directions The pressure to the sides left and right is the same as the pressure going down And so what happens here is this dam is not a submerged, uh horizontal plate It's actually vertically standing and so you have water that's exerting a force on the side of this dam And we're interested in how strong is that force? Whoops How strong is that force? All right, and so it turns out we can actually use we can use Uh integration to help us calculate the hydrostatic force against this dam And so let's see how that's going to work So to begin with like with pretty much every, uh integration problem that we try to do We had to set up some type of coordinate system. We have this dam in front of us, but there's no There's no variables in play. There's no x. There's no y. We have to define them So what we're going to do is we're going to place the origin of our geometry at the surface of the water And we're going to orient the x-axis in a downward direction That kind of like a downward dog pose, right? So we're going to place x equals zero right here at the top And we're going to orient, uh the the positive x-axis downward like so And this is a little bit that this is just a matter of flavor, right? Where do you choose your your x coordinates because it does make sense There there is a reason where you might take x actually to be the bottom Of the dam You could put x equals zero down here You could get bottom of the dam that way x is actually measuring the water level Uh, and you could calculate the hydrostatic force in that way An alternative strategy actually could be to set x equals zero at the top of the dam and then go downward Uh advantages of doing this and the other approach is that these are these aren't going to depend on the water level Which could change with the season, right? Um, and so So so like a engineer who works at this at this dam would probably set it up using one of those two things Um for us though, it's a little bit easier to set up these integrals if you take the the surface here Because if you set x equals zero to be the surface of the water Then we're going to see that x is actually equal to the depth or I should say the depth Is equal to x for this, right? How deep you go on the water is going to depend on your x coordinates So there's going to be a simplicity of doing that here All right, uh, let's see and so with the with any specific depth We can pick an arbitrary x value and look at a typical cross section, right? If we were to slice If we were to slice along the x axis a certain amount, we're going to have these little these little problems Little baby problems that we have to worry about right? What's the force of this at this rectangle? What's the force of this rectangle? What's the force of this rectangle? Etc and so the idea behind this is we're going to make an assumption. We're going to assume We're going to assume that uh, that basically all of the area is centralized You know that essentially I like to say we're going to assume that this is an easy problem That's that's how I like to think of that We're going to assume that all of this is sort of center centralized right here. All of the area is here So what we've seen what we saw before is that the pressure the pressure of a A pressure of the hydrostatic pressure That's the word I'm looking for is going to equal the density of the water Times the acceleration due to gravity times the depth right here And so by our assumption, but when I say easy what I mean is assume all of the pressure is the same throughout assume uniform Assume uniform pressure Throughout this problem. So when you have a cross section, we're going to assume that the entire pressure is the exact same Even though there is some width to it as these things are going to be delta x thick Um, we it's not a it's not a crude assumption to assume that it's the exact same Pressure throughout the whole thing. So our pressure in this case because again, we're working with the scientific units We're going to get 1,000 kilograms per meters cubed for water Uh, acceleration due to gravity is 9.8 and then we're going to get x So we get this 98 hundred For the pressure. That's really nice But force remember Force is pressure times area. So we have to also determine the area of the cross section Now, this is the part like I said, this pressure is going to be the exact same for every single problem. Um, Basically for scientific problems, this is going to be 9800 x Or for for british-american units, it'll be 62.5 x now the x of course could change It depends on your coordinate system. But if you place x equals zero at the surface of the water You're always going to get 9800 x right here. So that lack of variability is very comforting for students the area problem Well, what's that going to be? Well, if you'll notice our cross section is a rectangle. So its area is going to be length times width The the width of this guy is going to be dx Right here. That's how thick it is. And so the length is really what's suspect here And that is the part that this is basically the only part of our hydrostatic problem That we have to put really much mental strain in how do we determine the the width the length of this thing? Excuse me. The length is going to be this part right here Now one thing we came recognize is that no matter where we are On the dam here, you're always going to have at least 30 meters to your length Right. So like this portion right here is going to be 30 meters But then there's a portion right here. Let's call it little a And then my symmetry you're going to get the same little a over here So what we're going to observe is that the length is equal to 30 plus 2a So if we can ascertain this a value We can get the length and then we could we can go from there And again our width is actually dx. This is going to be the variable which we integrate with respect to So we have to determine a The the variable a adds a function of x and we're going to do a similar triangle type argument here So look at the following we have the entire dam shape Which is going to have It's 30 or it's 20 meters tall. Excuse me And then on the top it's 50 meters across the whole thing But we lose 30 meters in the middle and so there's 20 meters left for the sides Like so and then my symmetry we see that each side would be worth 10 right there So you're going to get this triangle which is 20 by 10 And then if we look at a smaller version of this triangle right here You're going to have this little triangle Which corresponds with our cross section you have a right here. What is this value on the side? It's tempting to say that this value on the side is x, but that's not quite right x is the x is the distance from the surface of the water Down Uh to the top of this little triangle. So the remaining portion would actually be Uh, we have to subtract this and again, it's tempting to say 20 minus x But notice x doesn't go from the entire top. It actually goes four meters down So actually the height of the water Which is in this case 16 20 minus 4 that is what we're going to take away We're going to take away from 16x there And so we get the other side of our triangle is going to be 16 minus x And so this is where the height of the water does come into play So we can see here that if you compare the ratios a over 16x compares to 10 over 20 or if you reduce that fraction one half Uh, and so times both sides by 16 minus x you're going to get a equals one half 16 minus x you could distribute the one half through if you want to but remembering that We actually don't want a we want to a If you plug that in there, you're going to get that the length is equal to 30 plus 16 minus x therefore we get 46 minus x as the length of of of the rectangles And so placing those in right here We see that the the area of each rectangle was going to look like 46 minus x times dx and so by putting these together the hydrostatic force Of the water against this dam is going to be pressure times area In which case we see the pressure is 9800 x times the area Is 46 minus x dx Uh, the last thing to determine are the bounds for x here. Where does x range? x can go anywhere from the top which is x equals zero all the way to the bottom Which is x equals 16 And which case we then get from zero to 16 right here So once this thing is set up, this is this is of course the important part here Once we get this thing set up the actual calculation isn't so bad whatsoever Um, let's actually take a look at what that would be. I would factor out the 9800 in front Uh distribute the x so you're going to get 46 x minus x squared dx still going from zero to 16 By the power rule We're going to get We're going to take x we're going to raise it to x squared over two Um, and then of course you could take half of 46 and get 23 there 23 x squared minus x cubed over three As you go from zero to 16 when you plug in the zero everything will vanish when you plug in the 16 not so much So you get 9800 times 23 multiply that by 16 squared minus 16 cubed over three This is the point where I'm just going to put this in a character to help me simplify this If you take 23 times 16 squared minus 16 third Um over three you're going to end up with 13,568 all over three times that by 9800 your exact answer would be 132966400 over three Put in some commas there you get uh, you know 132,966,400 over three and but again we we a rounded answer if we paid it to significant digits That's that's sort of a big thing in most science classes. I don't really care about that in a mathematics class But we're going to get that the force is going to be 4.3 So sorry 4.43 times 10 to the seventh newtons And we could upgrade this to like To kilo newtons or mega newtons if we wanted to but this is our answer and be aware that the actual arithmetic And the character's integral is not really what's of interest here. What is of interest is this integral right here? Why does this integral represent the hydrostatic force against the dam? That's the aspect I'd want you to get through this video and feel free to rewatch this and revisit this if you have some struggles Um with this or post some comments in the in the post a question in the comments below The point of these type of questions to understand why does this integral give us the calculation? Because it's a trivial use of calculation to actually find the answer Why does that integral give it to us? And we'll see some more examples to help us with deeper understanding of this And so look for those videos now