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# Show how to approximate the required work by a Riemann sum. Then express the work as an integral and evaluate it.A circular swimming pool has a diameter of 24 ft, the sides are 5 ft high, and the depth of the water is 4 ft. How much work is required to pump all of the water out over the side? (Use the fact that water weighs $62.5 lb/ft^3$.)

## $3.4 \times 10^{5}$

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so for things. Look at the trail to treat this problem. We have a circle of cool like a cylinder. The amulet only four means the radius is 12 The height, the pool or say the sizes five feet and the depth of the water is four feet. Is it for me? So they I think we slice the water, you know, very, very thin piece we call the ball of the thing paid spy down the B which is equals to the area attempts The area terms the thickness of this water and that's fucking overvalue. It's going to be won 44 6 And what's the force Forces value multiplied, by the way per unit. So one of the weight during the bottom is 60 to 1 bike. This is given its calling, though. Jeanne so f equals two Delevingne comes g. So that's 9000 high that X. So now we're gonna setting up over Raymond's son, which is limited, um, of assassination that endlessly affinity with some over I from zero to in 9000 high that x times exp I next presents the height 40 I slice. So basically it's here. We're slicing this This fight buy imports, and each part has the thickness exciting. That's the things for each part. So there's a room itself and to calculate the stream and some were gonna convert it to a river into girl the Riemann Integral, corresponding with women. So I'm going to set it up by zero before because force the depth of the water in ah, our functions right here. 9000 pie times, five minutes. Eggs B s this recall We're integrating over the water and the remain height is the Toto the total, um, the side both stylings of the pool minus over variable X. So that's over women's go. And, uh, What, integrating this. Anyone gonna have 9000 pie times two. So which is roughly 3.4 times 10 to the bike. That's a total work with it gonna do for this problem. Okay,

University of Illinois at Urbana-Champaign

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