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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.
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Calculus 2 / BC
Further Applications of Integration
Applications to Physics and Engineering
Applications of Integration
Oregon State University
Harvey Mudd College
University of Michigan - Ann Arbor
A vertical plate is submer…
$3-11$ A vertical plate is…
and this problem we're given that there is a place on Mercer water and nowhere has to find a force acting on one side of this plate. Let's assume not, um, this water level or the surface is our origin so that his X equals zero level. This would be executed street level and this final displayed it will be at X Eco's 11 feet level. Now we know that force is equal to pressure times area and pressures Eagle throw times, g times that 10 times area. And, um, we can write this one as under times D that times A or Linda is the specific way, which is you could throw times g. So we need to then find, uh, Linda will be already to know that for water that is sick up to 62.5 pounds for future Cube Now we don't need to find the that nd area. Now let's take this, Um, let's take this strip and we know that that is to feed. Let's assume that there is this tin strip that were interested in what a thickness D X. And let's say that if you were to some, all this 10 strips up along this edge. Then we would get the total force. Acting on that sort of area off distance trip would be two times DX. And if Roberto again integrate this area along this edge, we would find the total area none depth. Let's say that we're measuring the depth from the surface here. So let's assume that depth this age. Sorry, ex. Now let's play everything. And let's say that, um, force is done or infanticidal force. Acting on this infant is Immel. Thin strip is an equal to l thought that a 62.5 months fly by, um X times to the ex Effie, integrate this or Russ first, right? This one is a reminder. Some limit. As in ghost infinity, we have summation over I from one to end. Oh, I don't know what is 62.5 Times Expo. I multiplied by two times Delta X and we can actually call work that interim Interval and write this one as F force Total Force is equal to inter go from 32 11 what? 3 to 11? Because the this age starts at three and then we have this other edge Look I did that. It's 11 level. So that's why our limits for interview live from 3 to 11. We have 62.5 x to the X. We can let this one us two times 62.5 days is just a constant. We would get, uh, ex skirt over to where does changes between tree and 11. So this forces and equal to 125 that is to tempt 62.5 months what by 11 scarred over two minus three skirt over two. We didn't find the answer as 7000 comes.
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