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A group of engineers is building a parabolic satellite dish whose shape will be formed by rotating the curve $ y = ax^2 $ about the y-axis. If the dish is to have a 10-ft diameter and a maximum depth of 2 ft, find the value of $ a $ and the surface area of the dish.

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Calculus 2 / BC

Chapter 8

Further Applications of Integration

Section 2

Area of a Surface of Revolution

Applications of Integration

Baylor University

University of Michigan - Ann Arbor

Idaho State University

Boston College

Lectures

09:04

A group of engineers is bu…

09:13

05:47

Surface Area A satellite s…

02:26

02:38

An engineer designs a sate…

01:05

A satellite dish in the sh…

Okay, so section I want Teo problem 30. This is the problem asking to figure out the surface of your obtained by rotating the problem about why axes? So let's first organize what's what information is given by the problem. So given well, given that we have a problem going through an origin since my X axis and this is my wife axes and wear given the curve, wiII goes a X, we don't know what is, but it's going to look like something like this. Is the problem going through the origin? I know it's going to dorne because I've plugged zero in axe. I get a times zero equals y So the curve goes through 00 the origin okay. And by rotating this curve about why axes like this, we're going to get a dish looking like this. So I like to say, let's say this is the middle point. So if you look at it from the Bob, this is a perfect circle, and this is the bottom of it. Thank you. This is the bottom of it. And the bottom touches the origin here. Okay. And then he says the diameter is 10. So di you're meaner. If you look at it after circle in the diameter, here is 10. Okay, and max depth off this station, it's kind of look like a bull, but yeah, depth of this dish. So from the top to the bottom from top to the bottom, this length or depth off the dishes too. OK, so these numbers correspond to on the graph from some here from here to here. If I draw a straight line from this Lynch of this red line is 10. And this curve is a symmetric about why access? So if I drop a line from here straight down to the x X axis, I should get this should be fine because that corresponds to the radius of the circle off the surface of the circle. Right? And and Death Max steps to corresponds to Let's change color here. This point should be why goes to Okay, So that's two given information and the problems asking, asking us to find two things. So find first find the value of a A here. Why cause a scored find value away that describing this curve and it's also asking us to find the surface area off the dish. Okay, so given this information, let's try to find a one into. So first find a way. Well, if you go back to the graph, we just draw the shows that this line goes to X equals five. And what it goes to this point here is five too, right? So let's use that information to find out what that is. So why equals a X squared goes through point five, Tim. Therefore, when Why goes to I have X equals fine, that's what it means and X squared. So it's five square and that just for a then I have a equals two over 25 and we got dancing. That's your answer. This's the value of a so next. Oops. Next part two. Find surface area obtained by obtained. Bye. Rotating. Well, we found out the curve. Yeah, is actually why equals to over 25 x square, right? Because we found what the aids. So rotating this kerb, if lex equals to over 25 x squared nor a rotating this curve about, uh, y axis. So that's what we want to figure out. Why access. All right, now, remember the formula relearning this section says surface area biggest equals integral Off to my ex times Scare middle one plus determinative Off the curve. Fullbacks squared DX. This's the formula we learned in dissection, Clay. Now we found out what the f of X is so weakened our compute this interval and we can't We will find out what the SS. All right. So continuing from the previous screen surface area s equals integral of to my ex times. Square it off. One plus f o x was to over 25 x squared. This is our effort, Becks, and we want to take a dividend. Oh, bit. And then he's going to square. Hey. Okay, DX Also, we want to find out to Seoul for X. Want to find out what the integral integration bouncers. We need some numbers from from here here. Now, let's go back to the graph. We are going to integrate, um, about X from 0 to 5. The costas d a radius of the circle. Okay, so the integration bomb here should be We are going to integrate this from zero 25 So that's a valid Is angel okay to buy X, scram it of okay. Won't plus Let's take a generative of this first. So if I take a devoted to over 25 x squared, I get for over 25 x and we're going to square this t x teo to find two by x times square root of one plus Ah for over 25 x squared 625 in the bottom 16 and uniter X squared T X good tio 252 by X then I can't. So you've itis like there's got a computer inside the square it so I have safe 125 plus 16 x squared over 625 d x. Hey, let's go to that way. Okay, but the six and 25 was 25 squared, So all right, 0 to 5. I have to pi x over 25 and then square of 625 plus 16. Exquisite because this equals square of new mirror over squared of denominator right in the square of denominator just becomes 25. So this whole thing to buy over 25 this whole thing become just a constant. And I have This is the new matter we have here still in the red again. A magical sign D x now we need to use This is little complicated, but remember, we need to use the substitution technique to solve this interval. So let's put thiss Radic and as you So now put you equals 625 plus 16 x squared. Then I have to you over DX equals 32. Thanks. I just took a David of this substitution and then I have by rearranging it I get X d. X equals Do you over 32. I'm just I just rearranged us, but just like a regular algebra, then we can rewrite this integral with the substitution with hell. So then, ah, sequels I have Let's worry about the bounds later. For now, I want to rewrite this with you with so two pi over 25 not notice. I have X here and d x here. So there's the same us to buy over 25 times this radical x d x ray. So inside a radical becomes to be just you. Because that's what I said as you to be. U equals this radical and instead of x t x, I have x d X equals d over 30 too. So do you. Over 32. 30 to Okay, this is a little lettuce. Looks much single, and this is what we're going to solve. But list war about this about now. So when we have this equation terms ex, I got 0 to 0 and five. So when X equals zero, based on the substitution, I'll get If I plug zero into X, I get 625. So this here should be 625 because now this whole thing is in terms of you. Now X equals five. When X equals five, I get u equals you just plug it in. Plug fiving to here I get you equals 1,000 25. So this is the new integral that we want to solve for All right, we're almost there. So surface area is now integral from 625 to 1,025 and to buy over 25 square of you do you over 32. Okay, well, decision. Dear times one over 32 is sitting in front of the year, so let's combine all the constant and take outside the integral. Then I'll have, uh 400 high over 400. Why did I get 400? Because see to counsel with 32. Then I get 16. 16 times 25. I get 400. So pi over 400 an integral to 65 once those 25 square it of you Tiu Hey! Yeah, Okay. And pi over 400 times. Two over three times. You raised too three over too. And we're gonna validate this on 6 to 5 to 1,000 25. Let's go that way on DH, we just plug in. All these bounce in here, eh? In any subject so well too. But, Klink, you can clean up a little bit here. There's 200 so 200 times three is 600. So I get pie over 600. Yeah, and I plug in 1025 into X rays to three over two minus 625. Raise too. Me over too. And if you computers in a car collector, you should get something. I should do the approximation Sign approximately something like 90 point Dio 1193 and dust the valiant dust surface area off the dish. But this is as his approximately 91 90.11 93 square feet. And that's the answer

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