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Find the area of the surface generated when the curve in Exercise 34 is revolved about the $x$ -axis.

$S=\frac{\pi}{32}[18 \sqrt{5}-\ln (2+\sqrt{5})]$

Calculus 1 / AB

Calculus 2 / BC

Chapter 7

PRINCIPLES OF INTEGRAL EVALUATION

Section 4

Trigonometric Substitutions

Integrals

Integration

Integration Techniques

Trig Integrals

Trig Substitution

Missouri State University

Oregon State University

University of Michigan - Ann Arbor

University of Nottingham

Lectures

02:15

In mathematics, a trigonom…

01:49

In mathematics, trigonomet…

05:28

Find the area of the surfa…

02:04

Computing surface areas Fi…

02:19

01:38

01:17

08:44

01:46

04:48

01:41

The given curve is rotated…

02:01

01:27

Find the exact area of the…

04:19

Finding the Area of a Surf…

01:11

Surface area Find the area…

02:44

02:11

05:17

06:07

01:58

01:55

01:32

all right, because we're rotating around the Y axis. Our surface area formula looks like follows with thanks as our radius square this derivative anything great against ey. All right. No, um, we have why, in terms of X, but we need X in terms of why. Fortunately, this is just a couple steps of algebra, and we get 1/4. Why? Plus 1/4. We already have the bounds for why it ranges from three to 15. And also, since there's ah, since this is a line, we know that it doesn't cross over itself, which is important because we couldn't just use one integral, You know, if the if, why did something, like, you know, if we had something like, um, actually Okay, so we're rotating around. Why? So that wouldn't be a problem if we had something like this. We were rotating around, and it kind of overlapped itself, you know? Then we kind of need a We need a breakdown. The inter girls a little differently, but we just have a line. Okay, so there's no overlap. You know, we can return to the main problem. Fortunately, the derivative dx dy Why is gonna be pretty straightforward again. This is a line. So dx dy y is just equal to 1/4. Okay. And by the way, one plus 1/4 squared is just Route 17/16. Okay? And the reason is the first term, the one is 16/16 in the second term is 1/16. So we get 17/16 and that's just rude. 17/4. Okay. Um, So what this means is that our surface area formula, in fact, I'll just swap it out here, ends up being 1/4. Why? Plus 1/4 times? Um, I don't need to go top and bottom. Just Route 17 over four. See why? And we're going from 3 to 15. Good. No, to make your problem tad easier. We can sort of bring this constant out, all right? And we get if I times Route 17 over to and we're gonna know and then we are going to do well, we basically air just integrating this part, right? So I'm going to kind of jump ahead because this is just an application of the power rule, but we should end up with is in a greeting. Why squared times one a plus 1/4 times. Why? Evaluating from 3 to 15? What you should get doing this out is well, you end up with 1/8 times 15 square. That's to 25 here unless 1/4 times times 15 1/8 times. Nine. Okay, and there's a little bit of number crunching, but basically, at the end of the day, this is 30 around 17. It's not very good. 17 will fix that up to equals 15 pie times, Route 17 and that is your final answer.

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