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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.

$ x = \ln (2y + 1) $ , $ 0 \le y \le 1 $

a) $$

=2 \pi \int_{0}^{1} \ln (2 y+1) \sqrt{1+\frac{4}{(2 y+1)^{2}}} d y

$$

b) $$

x \text { -axis: } 4.2583, \quad y \text { -axis : } 5.6053

$$

Applications of Integration

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Campbell University

Harvey Mudd College

University of Michigan - Ann Arbor

So you have. You have this curve but is described by eggs is equal to natural libido off two white plus one. So it is easier to see these our X as a function of way. So no sleep. The access is we do usually no discussion on bond. It wasn't like that for For expose toe zero, we're gonna have, uh, for y equals to zero program. So zero plus one. So that's one zero. So he's gonna have the 00.0 and then well, we have this card For what? Between zero and one. Yeah. So this curve is like a natural with curved, but, uh, this place, uh, so by minus one half should be, like, over there was one house. So So the curve is gonna look something like the eso when the compute Ah, the area of Well, what happens if we were talking about the x axis? So we obtain All right? Like this, uh, curved cone. Something like this. We're data about the X axes, but so that we do to come kicked, huh? This here since here. This the exact assist us our y axis waas usually place so the way to complete. That would be the integral from zero upto one off office. The wife. But this is issue control the length of the differential. Well, this well, there is differential off length of the curves that is one plus if prime squared. Where with describe with these functions being therefore what? So that our f prime the rating over that if prime physical tool the movies is one over Do wife does one and then by the Adriatic camps too. So the b squared? Yeah. If prime squared, she goto four square over two. Plus, the wife is one square. Um, so, uh, so that, um also here are my battle here three x for completing the would be Why? Because we have Thio were treatable. Codex, this here's why. So this is the area area overall. Okay, off these were taking about is about X, which it is this interval. They dropped from zero after one. Why times the square root one plus this term for over. So why plus one squared the n d y onda. Uh, if we rotate about ex about the the Y axis Yeah, would be so here. The only difference would be that here we have a four. Why, that is. Oh, this is to fight on that. So, for these, uh, where the global why we would obtain something, Something like this. Hold on like this. Her in shape Like, uh, one like official, something like these. Um, So So this one has area is one his area to buy in zero from 0 to 1 off. Why? Thanks, Uh, that Wow. Plus a crime boy. The ideal surf. The prime squared. So this interval is is equal to by times go from 0 to 1 off natural relating to y last one times this quote one plus four over to why was one squared? And why on a lot these numbers you about your favored clear this number is, um mhm five. Approximately 5.6 53 from this one. Over here is son 4.25 He three. So this is the Syria. Well, about uh huh. Rotating off the surface of pain by retaining about. Okay, occasion about three x axis. The x axis is this one. So it is, uh, these cheeks. Uh huh. And then we're taking about both y You have this area. So this one is area, uh, by we're taking this curved rotation about about the white axis mixed up

University of Colorado at Boulder

Applications of Integration