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A vertical plate is submerged (or partially submerged) in water and has the indicated shape. Explain how to approximate the hydrostatic force against one side of the plate by a Riemann sum.Then express the force as an integral and evaluate it.
$\delta\left(\frac{128}{9}\right) \approx 889$ lb $\quad[\delta \approx 62.5]$
Calculus 2 / BC
Chapter 8
Further Applications of Integration
Section 3
Applications to Physics and Engineering
Applications of Integration
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in this problem, we're given that there is an object partial ISAF, mercy in water and then whereas to find total force, acting on one side of this object. And we know that force is equal to pressure times area and that is wrote OMG, Touch, Desire Death Pubs area And we can write this one a specific wait time step times area and for water. We know what specific ways that a £62.5 per feet cube Now we know this. We don't need to determine detect and the area in order to calculate the total force for this object. You know, I said that we have Let's zoom into this object. This is our object and some port. If it has some immersed in water, what's that out? This is the symmetric plane, and let's assume this point to be the origin. So at this point right here, if this is our symmetry, access or took, Alexis should be origin. And let's assume the sexes to be X and this to B y. So let's try and record and help this point to be 00 Now let's say that we have a 10 strip infanticidal element unless it ever interested in area. There's unless assume thickness off this element to be D Y or the height of it as doing so. The total area would be this Ling l terms de Juan. You have since ah, the shape of this object along the height changes This l will change as well and we want to know we would like to find a function representation for this l in terms of why How can we do that? Well, if you were to find any question off this ah, lying then we could find an expression for that else. Which means that we first need to find an expression for that blue line so that we could cackle it d area. Okay, if this is our origin and if the total length is eight feet And if it is Semitic me know that this part is four feet long and it senses the origin, the cornice off this point will be four and zero. Now we would like to find accordance off the sport as well. We know that total This is from this origin up to this point in one direction ist three years given And since this total listens or total length here is four. And this summer take it means that it is still here. So the coordinates will read into three. So now we're trying to ride any question for this line, and we know the accordance off in points and ah, our starting point and the point we know that line will be for y minus one on his eagle m times x minus X Not them will be the difference between wise and access. So it will be then three minus zero, divided by two months for that of B negative. Three over. To combining these, we can wreck the question of the light as why minus I'm using this point. It doesn't matter which point be used. So y minus zero is equal today to 3.2 X minus four. Now, since we assume they height off this infinitesimal element to be d y, let's write everything as a functional fire in terms of why Celeste, leave Max alone. Excell then be equal to four minus two are over three. All right, so now we know that dealing on one side and four minus two. Why over three If that is the case and if this ship is systematic with respect to this vertical access or the Y axis, it means that link on this side is also four months to y three. So this area off this green, um, region done infinitesimal area would be two times four minus two. Why over three months fly by height D y. Now let's find an expression for that. D It would be if we assumed I. If you assume I want to be positive, Amy upwards pointing outwards. And if this is our origin, don't forget. Whenever we're complaining, we're measuring pressure. We always take the surface to be a reference. So if this total distance from surface to the bottom of the subject is too, it means that the depth is a function of all. Why will be to my missus? Why? So that's right. He has two minds. Why must, like all those would give us deep force for this infant decimal objective eye. In order to find a total force, we would need some all those forces up Sword force would be limit as an ghostie Infinity summation off forces acting on those objects up so I from want and he have ah, specific way. 62.5 times the depth which has to minus y months fly by the area two times. So that would be Y two times four minus two. What I did wind front three times. Delta one and we are actually could write this one as integral physical internal knowing to decide on the limits off this integral we know not the object. This is our origin. And this is between the origin and the water surface as two feet. So it'll be from 20 to 2 62.5 times two minus. Why, I times two times four minus two. Why, sorry? That would be one, too. I over three times t y that is 1 25 is just accustomed at a 62.5 multiplied by two internal 0 to 2 to minus y. What's black by four minus two. Why or three d Y? And if you calculate descent trickle, we see that the force acting on this object is then equal to 888.88 pounds, or we can let. This one is 889 pounds.
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