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Problem 27 Easy Difficulty

If the region $ \Re = \{(x, y) \mid x \ge 1 , 0 \le y \le \frac{1}{x}\} $ is rotated about the x-axis, the volume of the resulting solid is finite (see Exercise 7.8.63). Show that the surface area is infinite. (The surface is shown in the figure and is known as Gabriel's horn.)

Answer

$$2 \pi \int_{1}^{\infty} \frac{1}{x} d x$$

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Video Transcript

sahafa of the following were Yeah, function, uh, for the region, but he's described as the following function to have We'll hear the function one over X. It sounds like the way up to community Small in that direction is the point of one. Excellent. This is the graph off. Why is he going to want other ex? Um, so it happens that we if you rotate his region around the x axis, we're gonna have obtained like a hornet, Like something but God. So we'll, uh, consider this region. But he's, uh, points up. Well, um, that is described by Well, we have eggs, beer on one, and then the points that are rely on this curve, why is he will toe one over X. So this collection of points points Exxon enough white because one or X. So the collections of points out eggs one of our x or Y on Dana. All such that our X is bigger than one. Don't. Yeah. Oh, you're smaller than a rico. The you consider the volume we know that region are taking updating our reputation about X. This has finite border. However, if you consider the area off, this expects the surface melting by rotating about the exact is more on the that area is gonna be equal toe or to buy the interval from one up to infinity Off function on the bricks times the square root One class as we're here we should have Yes, I'm squared, but for dysfunction for if affects being equal to one over X if crime she goes to minus one of her x squared so that this fine square is gonna be Uh huh. He's squared with friends squirt vehicle to one over X to the fourth thanks to the fourth power. Um, next before, so you would have something like this. The X and we have is that this number goes to infinity. You saw it does not come British. So does not converse on that. Well, the reason well one can justify that is because, Well, these this is bounded by below are by might be seem to go being in our off just one of her eggs because these number squared off one plus why were excellent work for X better than one. This number is greater that one that if we multiply sound function by a number that is your number one here. Nine One way Will will obtain something bigger. So I well, these one that is that is a lower bound. This integral is already Wow Is he drove gold still Take your store is not It's not Finally, finally so that this interval that should be bigger is also going to infinity. And so he's not planning on thought. That is the products. So for others, because we all love a region with finite volume. All right, this nice vision are girl, you know, pain by by what they're doing that that hornet this region are has finite volume by night. Yeah. Mowlam the Internet Huge service self area. So that you would Yeah, we have these region. And then you Would you like? You would like to buy some paint to To paint all the surface area You go broke You broke because you need infinite pain. You need the The region is the finance woman

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