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Sketch the region enclosed by the given curves. Decide whether to integrate with respect to $ x $ and $ y $. Draw a typical approximating rectangle and label its height and width. Then find the area of the region.

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

$=\ln 2-\frac{1}{2} $

Applications of Integration

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Missouri State University

Oregon State University

Harvey Mudd College

University of Nottingham

All right. So verse every draw X y excess and go the forest. First function, like owes one over X Terry's one. What? Okay, So this is why you go to while Rex and we know except goes to Ah, this vertical line here. And what, over x square? Um, so, yeah, you know, X square acts is bigger than one. It goes faster, ex Clarice. Bigger than acts. So the reciprocal of X square of a Latin will be less so she do something like this, All right, Because this point, um, or lower acts corresponding to where half at this point here. The access codes to of it. Plug in May, we called one fourth a quarter. Right? So that's why this curve is lower is that while we're act square and up her wise power x okay and the enclosed regions right there get, um, and everything is represented. That ax. So we decide. Integrate respect to X star treks on. Then we draw a typical approximating rectangle. So say the rectangle is right here. I know the zoom out. It will be something like this. This is approximate rectangle near the callousness Delta X and the height will be the upper curve minus the lower Kurt is one over X I minus one over X eyes. Where? Right. So, um or less Jarvis to find Ariel, this region and then by this formula we know area the egos through the into a roll. Um, with respect to X. So right now our jack's here, and then they know our X goes from one to two because the intersection here is one one, and, uh, they stopped at X supposed to. So it goes from one to two, and the integral inside it will be well over X Will be this without the subscript. Uh, well, our X minus one over x square. Then we need to find Auntie. You're okay about this? You know what? Or ex Ante Dios, you allow our access log X. So, Sophie, Log X and, uh, no Aunt Aunt Edie narrative for negative One X square. It's just while Rex. So this whole thing Nandi directive off this whole thing will be log X plus one over X. And I would take the boundary matter whose X equals two and actually close to one. So this is very close to love too, Because one or two minus love one, which is zero plus What right? This area. There will be love too. One half minus one is minus one. Yeah.