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