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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.

## $\frac{2}{3} \delta a h^{2}$

#### Topics

Applications of Integration

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##### Kristen K.

University of Michigan - Ann Arbor

##### Michael J.

Idaho State University

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### Video Transcript

in this trouble were given that there's an object with a trip hazard, a lacrosse action and I were testifying. Total Force Acting London's object we don't force is equal to pressure times area. And that is you go to density times gravitational exploration times Did Beth measure from the water surface times the area? Now we know what densities we confined or we know what gravitational exodus, ingenious and if he assume or origin to be here and if he let's use a different color if he said that this is the origin and if you send this to be the eggs and this to be the Y axis and since we know that doctors measure from the water surfaced since that is our reference, we could say that this is that it means that by setting up accordance is someone origin we could find. The depth now we need to do is to find an expression for the area. Now let's zoom into this trip is a little object. This is our object and we know that depending on the depth, why the with office object changes, so the area off distant stripped will be this would which is a function of why w If I want to blind by de Wine. All right now we need to find an expression for that with off this object as a functional for How can we do that? Well, we can do that by writing any question. Who's writing the question for this line that connects this point with that point? So now let's write the coordinates off those points, since we assume our according or origin to be here so that it is years your recording itself, the upper point would then be a comma zero since photo link. This to a decided would be a this would be a and the origin or recordings off. The bottom point would be a over to comma H since total height off this object this age and since decide is able to turn this, say we're too. Now we're trying to Dan, right? Any question of the line would add endpoints, a comma zero and a or two. Come on, H b. Not any question off the line will be up for y minus. Why not busy? Go to Slope times exploits. Excellent. Let's use this point as are given points or why minus zero will be equal to M. The slope is the difference between wise divided by different between access or out of the H minus zero, divided by a over to minus a multiplied by ex minds. A. From this, we see that the question off the line is then eight times why plus two age Times X is equal to eight times a sense we're trying to write the with as a functional wirelessly ex alone X would be equal to a minus over to age times. Why, all right, Yeah, this is 1/2 of the word for the total. What would be w? Why would then be two times a minus over to age times? Why and the area would Gumby Inter go from Since the total height is h from zero h two times a minus our to age times? Why? What supplied by the height off this 10 stripped which is do you want? All right now let's play everything we know And in order to find total force, we would need to force we would need to some the forces up that is acting on this tin strip so we can write that as the Riemann some so force would be equal to limit, as in coach Infinity summation from Want to End. They would have ro times, g times death, which we assume origin to be here. And we assume that to be Why So why Times Area, which is two times that every why I got is two times a minus eight times why I do you mind bye to age multiplied by Delta y. So that is re Munson. Now, in order to calculate the total force, we could homer that into an inter gal's so D force would be internal from zero to age, since that is the, um that or total height off this object. Zero h He had ro times g times y times two times a minus a y over to h times D y to Rogie. Doors are constant. We could write this one as the two times right time she integral from zero h a. Y, minus a white skirt over to age times D y. That is equal to two row G Time's a Y squared over two minus a white cube over six age where why changes from zero H and quandary. Calculate Dez this part. We find out the total force in terms of two, given our variables s two times eight scribe age eight times wrote times G divided by three

#### Topics

Applications of Integration

##### Kristen K.

University of Michigan - Ann Arbor

##### Michael J.

Idaho State University

Lectures

Join Bootcamp