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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.

## $313,600 \mathrm{N}$

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Applications of Integration

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in this problem, we're given that there is an object with triangular course section. And now, whereas to find the force acting on runnin off this object, we know that force. Is he going to pressure times area and that is equal to road time. She times depth times the area. Now we know what drawers and we know what he is. Siro, It's 1000 kilogram per meter cube for water and G is 9.8 meters per second skirt now. I mean, it is German, but days and what a is Yeah, let's zoom into this object under sent what area? As, um less assume not be harness 10 sheet when a thickness d y, um So the area would be dealing l a times d by now the key or the idea is to write this l in terms off one. Now for that, we're going to assume that we have an origin located right here. So this is origin 00 dishes are XX ISS and this is our y anxious in order to find El in terms of why we would need to find the question off this very lane. So the coordinates off the end points are done since the height off this triangle's two. That would be zero comma, too. And this edge, this word text is located at three comma zero. So we're looking at that line with 10 points. Sirrah, come on. Two and three. Come, Sirrah. And the question that line would be why minus why not is equaled. M times X minus X not. Mm. Here is to minus zero. Divide by zero minus streets of difference between wise divide about difference between excess orders Niner three to any question will be then why minus Here is ICO to negative three x or T plus two. We want to write everything in terms of why. So let's leave ex alone and right at one of us. Axe is equal to three minus three over two. Why? Okay, now this means that if this is the origin this distance or one side, one end off or 1/2 off this link is a three month tour to one. But don't forget that a symmetric. So we have the same link on the opposite side. So the area this area off this tin sheet, I will say differential elements or D A. Will be two times three minus two. Why were, too terms tea? Wine? So what is? Since now, we found an expression for area B last leading to determine what, um that this If this is our war and if we assume this is the positive, why don't forget one measuring pressure. Be always measured, that studying from the surface. And if this total link is six meters, that has given problem statement that in terms of why would then be six minds want sort of total language? Still be four plus two? That would be six. So then the area or the force acting moments differential element This red block Pdf would be raw time Caesar 1000 times 9.8 times Ah, six minus wind that is Tibet. The month black by the differential area is two times three minus G. Wildwood, too times D y In order to find a total force, we could drive or we could some beat those elements up some all those forces up, and we would use a remote sums, a limit as a ghost. Impunity summation From one to end, he had 1000 times 9.8 attempts six minus y times two times three months to G A Y or two d y, or we could call murder at Inter Inter Galang right f s f now in to decide on the limits. And the limits will be between right here so that I will start from zero the first blue dot and it will go up to the second blue one. So that is between zero in 2000 terms 9.8 times two doors are constant. Times six minus y multiplied by this one. And we will get, um 18 minus stroke y plus three wife's card or two Do you want? So the first part is just a constant. We could write this one as 19,600 times inter go 0 to 18 minus 12 y plus t y skirt over to D Y. That would be 19,600 times 18 y minus 61 squared, plus white cube over three planted at zero and two, and we don't would find the total force as 313,600. Nugent's

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