Show how to approximate the required work by a Riemann sum. Then express the work as an integral and evaluate it.
A leaky 10-kg bucket is lifted from the ground to a height of 12 m at a constant speed with a rope that weighs 0.8 kg/m. Initially the bucket contains 36 kg of water, but the water leaks at a constant rate and finishes draining just as the bucket reaches the 12-m level. How much work is done?
Applications of Integration
Okay, so we're gonna calculate the work of this system. So the system consisting of 23 parts, the water is at the market, the rope and the buck itself. So the mass of the whole system peoples to m one cost them 2003 and know that I'm one of two. So this is Thea. Mask the water and two with the mess with the road. And those two masses are a variable off the tante and M three's the buckets, which is a constant. So sateen not equals two. The time that you're gonna pull out. Ho ho! What? The pockets with a constant speed Because the length is tell and say that we this because this speech is gonna pull up market sober. T not. It's the time that needs you. Pull up the system or pull up the bucket. Think of everything in the system so we can right out. I'm one and two with results on T buying warranty. Not recall that A mass the entire water inside a bucket to 36 So I'm one is 36 times one over tea. Over t not. And for the road is nine on the six times, one minus t o Pretty nuts and three's a constant and, uh, let's see and through equals two. Okay, that's the setting up. Now consider the work the work. Let's consider work that's that's done in a very small period of time. Like even a testable time said D T or we can see lt But guilty goes zero. But either way, so the work done in this very short time is, um, one plus on one team was sent to a T. Let's call it D W plus I'm three. Sense of gravity comes to velocity times T t That's over, immigrants. This party's over in the grounds. So how they wait, How do we track with a total work? What is there any girl from zero to see? Not recall. Tina's the time to pull the market in a constant speed, which equals two and one tea for us. And to have tea wasn't three country DT times. The exhibits constant. It's independent of tea, and we're plugging all those expressions here and the boys 10 times G b t. Not waas. 45 26 times g times bt not minus Haven't waited. Not so this part of the stuff half we cannot. I wouldn't know that c equals to shelf over. Be so wishing. Plus, sweetie not equals to 12. And plug in jail was to 9.8 made of her square Second. We're gonna have these results w equals two approximately 3800 57 Jules.