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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.

$F=9800\left(144 \sin ^{-1}(2 / 3)+\frac{176 \sqrt{5}}{3}-144\right) \approx 9.04 \times 10^{5} \mathrm{N}$

Applications of Integration

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Ralph G.

July 31, 2021

love your explanations! clear and concise!

Campbell University

University of Michigan - Ann Arbor

University of Nottingham

Idaho State University

in this trouble were given that there's an object with someone circular cross section of submerged in the water. Um, whereas to find a total force acting on, sign up this object now, since we know that there's a, um, object and the cross section Is this in my circle? You know, not then. If total diameters 12 total length is 12 and radius will be six meters and then the equation off, this will be X squared, plus y squared days six squared. That is a question of a circle. L assume not be have an origin that is located by at the center off the circles at a +00 Um, and from this, we could write X as a function of want as X is equal to skirted of 36 minus y squared. Now why do we wanna write X is a functional flying now less Take this object. Let's zoom into that. We have someone circle and some of bridge is somewheres in water. All right, now we want to find the force, and we know that force is equal to pressure times area on that is equal to row times g times, step times the area. No vino Roland G Row is 1000 kilograms per meter. Cute and me know that's G as in 9.8 meters per second squared. Now we want to find the BEP and the area. An area for that one would be the summation off the area's off the stain or infanticidal differential areas. So this 10 strips But since you're a distance trip has and thickness off the wine so we would need that total link right? And this total linked this link would be, oh, our inner. To find that we would need to know the relationship between X and Y. That is why the first need to drive this. All right, Now we know what X is a functional what is right. So we know it. Only one health X is equal discredited of 36 minus y scrape. But this is Sima takes incidents a circle. So he hopes we have the same exact one on the opposite side. That is also the spirit of 36 months. What's cried for a total area off this differential element D A is going to times 36 minus y scribed scare Rude. What's black by T. Why now We found this. We know this. We know that snow in determined deep that let's say that, um, this is the origin, as we said, and this is positive Why, right? If that is supposed to what now? Don't forget to be measured or pressure as a function off depth, and we always measure the depth starting from the surface. Okay, if the total length between surface Andy center or the origin is for it means that that as a function of why would then be four minus one now then the force acting on this infanticidal element d A would be d f that is infanticide, milk or differential force. And that would be wrote times G times D, which is four minus y times differential area that is two times curdled up 36 months. Why skirt, do I? So, in order to find total force acting on this subject and we would need to some those areas up so less than right total force as a reminder some that is limit as and go see Infinity summation for I want to end. We have ro times, he so that is 1000 times 9.8 times four minus one I'm to skirted of 36 once. Why I squared Tubbs Delta y will Death wagon is a sickness, and we can cover that into an integral run. Right? This one, US f Now we need to determine Decide on the limits off this inter go. And we know that the portion that is summers in water as four meters. So that'll be starting from zero up the surface dessert before he had 1000 Terps 9.8 times four minus y must let my two skirted of 36 months y squared do I and immolating this integral. We see it up on the force acting on this object if 9.4 times 10 to the fifth Neptune's.