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(a) Set up an integral for the area of the surface obtained by rotating the curve about (i) the x-axis and (ii) the y-axis.

(b) Use the numerical integration capability of a calculator to evaluate the surface areas correct to four decimal places.

$ y = \tan x $ , $ 0 \le x \le \frac{\pi}{3} $

(b) (i) 10.5017

(ii) 7.9353

Applications of Integration

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Saira K.

October 17, 2021

integration

Harvey Mudd College

Baylor University

University of Michigan - Ann Arbor

the following problem. You have, uh, of this curved? Yeah, in the x y plane. Because he's, uh why is he going to tangent Effects? Is this Why equals tangent off X way? Have that curve from X equals to zero up to, um what is yours Aware? Uh, Kennedy at my house. Somebody we have these one up to thirds. And so all this continues here. So you have to consider just this portion from zero of two by thirds When you want to do the following First of all this money, want to update about the the X axis? Correct. Would rotate that piece. This piece from zero up to buy thirds off. I didn't like that. You get something like that, Someone sort of like a trumpet. Something like that. So I want to compute. What is the area of this? Um, so with the x axis and then Well, look, we also like to rotate about the y axis. So you have the that curve here is the Y axis. This is our nice, wonderful curve on the Reato. Look, take on there. So we're gonna obtained, like, a a bowl, something like that. Were they so not to compute the area of these two services for this one. You know that, uh, by formula or the it's gonna be being to go from zero up to buy thirds. So after five thirds Yeah, off from, uh, if effects, uh is, um, all the length of the curve? Uh, just yes. Which of these The length of this car would be? School one. Bless crime squared. Well, that dance by. So that is the formula on the wall for this rotation about the X axis. And also this is that would be why this part here is why on then we rotate all the the X axis are what changes here. Is that here? Yeah, they dropped by from here. The bikers, uh, deep items. What appears here's would be just X consents. The X, um so for dysfunction for effects being called toe tangent fax over dessert. If effects of crime is equal to Seacon square, the roofs tangent this second so that, uh, over this thing to go Well, let's call these ones. Uh, mhm. So that is gonna be equal to yeah. To buy is two pi times the intro from zero up to by thirds off or function pungent fakes. Clams s. So these would be square root of one plus this square. So see, Camp Squared Square would be just speak into the fourth. See that square affects. Oh, that square is ableto see them to the fourth power. So you have seeking to the fourth? Yeah, it's the fourth is the X. So that is our first one we're taking about about X from being If you're a date about why do you have this? So that sort of it is, uh is she called to seven? One will be to fight. There was interest from 0 to 5 thirds. It affects Champs Square it off all the same. One last sentence. Uh, the fourth power that effects the fourth power. Yes. Way opposed to approximate that with our favorite calculator. Um, this is, um, 10 point 517 from this is seven 0.9 35 Okay, so those are the the approximation studies areas of revolution like that. Like that? Wow.

University of Colorado at Boulder

Applications of Integration