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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 thick cable, 60 ft long and weighing 180 lb, hangs from a winch on a crane. Compute in two different ways the work done if the winch winds up 25 ft of the cable.

(a) Follow the method of Example 4.

(b) Write a function for the weight of the remaining cable after $ x $ feet has been wound up by the winch. Estimate the amount of work done when the winch pulls up $ \Delta x ft $ of cable.

(a) $R=\frac{180}{60}=3$

(b) $3562.50 \mathrm{ft}-\mathrm{lb}$

Applications of Integration

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Oregon State University

Baylor University

University of Michigan - Ann Arbor

So first, let's figure out what is Thea adamant about? We're reading some. No, the way this 1 80 pounds and the cable is the length of the cable of 60 feet. So the racial are is one A or 60 equals 223 And, uh so that is the weaker link is three times the X, and that's gonna be over elements, you know, weariness up and Barbie with a calculator work they're requires to get a label. Gotta get a cable wanting up for 25 feats. And that's the wedding by two whores. The first part is, you know, the work to want it up and the second poor is second part is actually, um, the weight of the 30 but remaining feet of the table. Just think of it. You wanted something up. You have to Hey, some energy to to want it up. And, uh, you have to live out way have lived out the rest of the cable that is on wondering. Right? So it's actually, um, that one plus two. And for double one simple and easy equals to deserve to 25 3 times played by minus X, the eggs. That's the work. We used to wonder up the cable from 0 to 25 seats up all right. And that particles too. 970 37 points, creepy feeds guns. And second for is sitting part is the work we gonna pay for the remaining 35 feet at the table. And we know from the first way we know from our aid at our inter grants and three X three times down the eggs so poor and integral, which is to transform driven some to a ribbon into girl. It's gonna be 3 30 to 35 33 the eggs 333 times 20. But the ends, That's Theo, the length of the cable that has been lifted up. And that part is fixes number, which is people's too 70 welcomes 25 in the that's 2625 feet guns. So until the whole work able student, everyone wants it to, which equals two 3562 going to be bit guns. That's a total work, too. T make the job done. Okay,

University of Illinois at Urbana-Champaign

Applications of Integration