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00:06

Jeffery W.

If $f(x)=x+\sqrt{2-x}$ and $g(u)=u+\sqrt{2-u},$ is it true that $f=g ?$

01:02

Anshu R.

01:09

Felicia S.

00:56

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Okay, so this is a really interesting problem, because finding the surface area of the side of a cone, it's not something that I would consider to be common knowledge. It's not like the area of the circle. I would expect more people to be able to tell me what the area of the circle is, as opposed to the surface area of a come. And so just to be clear here, I only want the area of the side, so I don't really care about the bottom. I know the bottom is just gonna be pi r squared. And then I have a height, okay. And then the slant height is really the distance measured along the slant. So we call that not Yes, we'll call it. L like that. Okay. And again, I only want really the surface the side service right here. Okay. So what? What we can dio is we can parameter rise the surface and just plug it in the surface area formula. But I want to be a little careful here because R is the radius and l is the slain. So what I can think about if I actually draw this in space So if I let this be the Z axis right here than maybe I have my cone look, something like this goes around. So here's our up the top, and then I have a height. I'll just call the height age. I noticed that I'm not labeling my slant height, but we'll see how that comes in in a second. But what I really want to notice is that if I call this the radius of the circle so the radius of the circle is increasing the radius of the circle, or rather, the Z coordinate. If I look at this line, varies is a function of our by this slope h over our says he is really h over. Big are times little are now little ours. The variable. Okay, so that's just saying that Izzy increases the radius of these circles is going to increase by this factor. Okay, looks good. So now I can give a parameter ization of this Come right here. And so let's think about what that waas We actually just found it in the first example. But I do need to account for actually the fact that Z is burying a little bit differently of the radius. So I know that I'm gonna have s because Sinti s sign t But the Z coordinate is varying with the radius in this way. And so I'm really thinking about s is being the height right here. So Z is h over our, uh, h over started little our times Uh, h over big our times little are but my parameter s is really acting as my hyped right is really my Z coordinate. So really, my Z coordinate is going to be a church divided by our and then times s. And just to kind of make this clear, think about when s zero the z coordinate zero and when s is equal to are this then the ours they're going to cancel And I'm just left with h okay. And actually here. So Z is is my vertical coordinate. What I'm really thinking about s being here is the radio radi I of the circle. Okay, I mean, this book about that s is really the radius of the circle, so this is really like my s coordinate here. That's how the radio of circle are increasing the sea. All right, so now What do we dio? The most important thing we need to do is we need to find actually the magnitude of the cross product if you recall the formula. So the partial with respect s is just because Santi scientific and h over our okay, and then the partial with respect to t is minus s sign T esco, Sinti and zero. I guess the other thing I need to do is I need to actually write the domain of S and T. So s we said was the rate was the radi i of this circles as the cone increased. That's going from zero. The largest radius is our and then the angle T is going from zero to two pi. Okay, so that's gonna help when we actually do the double Integral. And so let's first find the cross product. Okay. And so recall that the x component I'm just looking here. So I'm going to get basically this negative of this times this so h s over are and then cosign t Then we'll like that out. And then this is going to be well, this minus this. But it's negative. Yeah. So h s over our sign t and then over here we have s co sign square T minus negative. So plus as Science Square T But that's just going to give me s all right. And now if we look at the magnitude of this cross product, then we just square all the components. But notice that this is just going to be this square times coastline squared, plus the square times side squared. So this is going to be the square root of H s over our all squared like that. Okay. And then plus, yes, squared like that. So what can we do? Well, I can factor out in S and Aiken notice that s is positive, and I can factor out. Let's see. Ah, one over are so I'll be left with s over are and then square root of h squared plus r squared. So just put that back under the square root and you'll see you'll get the same thing here. But look at the square root of h squared equals R squared. This is H coming down right here. Here's our We have a right triangle. Notice that h squared plus R squared is l squared so that means the square root of H square bazaar squared is just l So this is s over our times. L all right. So now to do the surface in your girl to find the surface area, that's just the double integral. So zero to two pi for tea and then 02 big are for s of this function s over our times l and then we have d s de t. But look what happened. So integrate with respect, that s We get s squared over two. So it becomes r squared over two, which is, in other words, just are over to when we divide by our. So this is 0 to 2 pi have just are over two times l and then, of course, DT. But then we integrate with respect to d t. We just end up multiplying by two pi. So what do we get? We get pie are which is exactly what we were supposed to get. So there you go. Now, if we wanted the surface area of the entire cone, we would just at the bottom, which would be pi r squared

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