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Use either a $ CAS $ or a table of integrals to f…

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Problem 23 Easy Difficulty

Use either a $ CAS $ or a table of integrals to find the exact area of the surface obtained by rotating the given curve about the x-axis.

$ y = \frac{1}{x} $ , $ 1 \le x \le 2 $

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Calculus 2 / BC

Calculus: Early Transcendentals

Chapter 8

Further Applications of Integration

Section 2

Area of a Surface of Revolution

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Video Transcript

and this question we're asked to find the surface area obtained by rotating this curve around the X axis and basically, okay, we're told that we can use the table of integral or a calculator, I'm going to use the calculator route but we ought to set up the integral first since we have our curve Y equals one of her ex. Yeah. By use of the power rule, we know our derivative is just going to be negative one over X. Where pretty straightforward. Now substituting all of this into our integral. We have the surface area will be equal to That. The integral from 1 to 2 will have to pie Yeah, Times Whenever X. But you look like that. And then we'll have the square root of one plus negative one over X squared the whole thing squared. And we can simplify this a little bit. So we'll have the surface area will be equal to one half times cheap. I over X. Times square of one plus one over X to the fourth power. And this is enough. This is enough simplification because when we plug this into Wolfram alpha, we're going to get the following times square of one plus one over X to the fourth. Yeah. Oops, not a p al sha mhm We'll get we'll get an exact answer of pi times root two -117 divided by four. Bye minus the ark. Cinch of one or the our sense of one plus the our sense of four. Yeah. Mm hmm. However, if you were to simplify this, if you were to do more simplifications. If you're to do more simplifications here, you would get a different answer involving law algorithm. You get a different answer involving logarithms and and other ark cinch things. Don't worry, these are the same. These are going to be the same because our cinch and the logarithms are basically the same. Yeah, yeah, yeah. So we'll use this answer. Yeah, that's how you do this question.

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