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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.

$9.8 \times 10^{3} \mathrm{N}$

Applications of Integration

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Campbell University

Oregon State University

Harvey Mudd College

University of Michigan - Ann Arbor

in this problem, We're given that there's an object. Um, order triangular cross section. And then where has to find a total force acting on the one end off this object. Now, let's assume not this symmetric point. These are origin. So this is our origin. That is 00 and that is the X axis. And this is deep. Why access says this is Semitic would suspected us. We're took a line. This distance would be one meters on. This is no with fuel meter. So this quarter and it's at this point would be then one and zero. Now we want to find the quarters off this point and I will explain why in a second now, since this length of all the says are the same, this angle a 60 degrees and dentist than 30 degrees. If this the length off the side is this one, then this length would be skirted off three meters. And since we issue in positive wide, the point outwards is means that the coordinates off this point will be zero and negative. Skirt off three. Now, the question is, why do we need that? Well, we know that forces a go to pressure times area. And that is you could throw Tongi terms that times area. What do we know? We know what drawers. We know that role. Is it good for water that is one kilogram per meter cube. You know, the G also is eagle to 9.8 meters per second squared, so we need to determine deep depth in the area. That area would be summation off areas off this 10 strips with sickness. Oh, you are So let's start at the sickness office would be D y and the area would be do our times the length off the size, right? And in order to find that link, we want to know, uh X in terms of why so the relationship between X and Y In order to do that, we're going to write any question off this line which line this very line. That's why we needed to find the coordinates off the end points. Now let's ride at me now that the question of the line will be up for one wins, one of his equals, M times exploits Exxon, and for one extent, we could use either of those points. Let's calculate the slope furs. Mm. Put B using given points, the difference between wise the wide by difference between excess a less turd zero minus native skirted or three divided by one minus zero. From this, we see that thing to slope is skirted of three. And the equation off the line from this isn't why minus zero. I'm using this point. Why? Minus There is equal to skirt or three times X minus one. Uh, since we said at the thickness is d Y. Let's write everything in terms of why so lessen the axle one sidle right X s. Why? Over skirted of three plus one. All right, if that is the case, it means that now let's zoom into this intense trip. Okay, we have this dance troupe, you know that the thickness of the Y and this line is special. And that is, that worked the line and we know that. Then this link on one side is why over skirted or three plus one, It means a sensitive Semitic. It should be same on the other side. So why over skirted or three plus one? It means that area D a so differential area off distance trip is then two times watch or skirt or three plus one times D y. Now what is that? We know that this is our origin this point and we know that when calculating pressure be measured in death from Betis's thought, this would be that Why? But as we assume, want to be positive pointing outwards. This will be negative. What? So now we know row G D and A using that, um, we can write the force acting on this 10 strip the infinite decimal force DF as road times g times negative wine times too. While we're skirted or three plus one times d y. In order to find total force, we get some the forces up. So that's why we're gonna use a remote some so that its limit as an goshi Infinity summation from one to end He have 1000 that is ro times G was 9.8 times negative. What I times two times why I do. I did skirted or three plus one d y, and we can actually write this one as in Tegel F What are the limits? Lemma Stan will be starting from the minimum office or lower about until the surface. So that'll be between native Skurdal 3 to 0 1000 times 9.8 times y times two times y or skirt of the three plus one d y. And calculating this in Tegel, we find the total force as 9800 new tricks, and that is the final and integral and the final result.