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# Let $L$ be the length of the curve $y = f(x)$ , $a \le x \le b$, where $f$ is positive and has a continuous derivative. Let $S_f$ be the surface area generated by rotating the curve about the x-axis. If $c$ is a positive constant, define $g(x) = f(x) + c$ and let $S_g$ be the corresponding surface area generated by the curve $y = g(x)$, $a \le x \le b$. Express $S_g$ in terms of $S_f$ and $L$.

## $$S_{g}=S_{f}+2 \pi c L$$

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Applications of Integration

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You seem very for a situation we have and look positive curve like this. This is the X axis. Somebody's the white. This is the girl. Why is he going toe off X on? Uh, we have some points here. Two points here, a B bond. Well, are assume that we compute the area off the surface of revolution. The service of revolution we're taking about the X axis. The fortune Give us something like these bureau Tate that can obtain something like that was so in the region. Not this region has some surface Mr Were very sequel to To find from Central from May have to be off if or fax is the way to compute. Gabriel, your service. Uh, the rotation some facilitation about. Come about X. Yeah. Thes times one plus the directive off our function squared, B x. So some very have done this for those two numbers is a this smf Andi data This curve has named, you know, So the length of a curve is on the length of that curve. Is this number being to go? Yes. Well, this component here crime spread? Yes. Is that yes. So the length of this curve with. We just need to offer May up to be Yes. The differential off of length off the curve, that is Yes. So now in this situation, um, but if you have, uh, we have our same portion. But, ah, if you consider some numbers, see some number here, see some positive number and we're gonna make the new function G faxes ableto our function ffx Plus that constant policy she would look like or like, We have a lot of fortune here on the disease. Self on the translator function would be like moving. Moving up. This is Jean moving up this function by C units. So the obtained service by rotating Jesus? Well, it would be also from a up to me Would be like this wider shape that has sort of like the same form. Us was the surfaces to vet, but But we got wider reviews. She would be like this shape. Yeah, here from the same points, A and B. So, yes, but SMG, which is there with the surface of revolution Service of revolution. Uh, how about XX? Is so this would be dangerous from a up to be off to five dams G I've seen to go on times square root of one plus g frying squared the X But well, if you translated from this equation translated G If that constant well, can we have the def prime is equal to G Fine. So that this part over here this is precisely, well the same. But considering that if so, that we have one plus, yes, frying squared. But is this confirmed here and then we'll f g s f fix. Let's see. So that s e g is gonna be ableto this integral to buy everything to go from zero from a activity off. These on the go can multiply this by that, then that by that and separate between intervals. So So that the service Yeah, of all the proceedings g about the expenses from were telling me about the exact is would be buying each other from that to be fast affects times the square root one plus a crime slurred p x and then plus C Mhm we'll see, is a constant so we can pull that out of the constant integral See, I'm stood by the interval for May off to be on these square with one plus here. Fine squared. Yes. So these are quantities that we know this part. This part This yes, Sabbath. So the surface the area of the surface of revolution by rotating, uh, in the original function, Yes. Survives precisely that number on the wall. There's a number of flies Have this constant to buy. Comes to see So for C positive number just be positive. Under these are the turnover. Here is the length of the curve. No, she saw you know that over here is in jail from hell to be off square with one plus if friend squared. Yes, it's that so This is hell. So let me so the surface of the area of the service of revolution. What the energy is the area off? Um, the regional function plus two five that was seated himself. Nobody's back in the area of this shop Modified surface

University of Colorado at Boulder

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Applications of Integration

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