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Formula 4 is valid only when $ f(x) \ge 0 $. Show that when $ f(x) $ is not necessarily positive, the formula for surface area becomes$$ S = \int_a^b 2 \pi \mid f(x) \mid \sqrt{1 + [f^{\prime}(x)]^2}\ dx $$

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$$S=\int_{a}^{b} 2 \pi|f(x)| \sqrt{1+\left[f^{\prime}(x)\right]^{2}} d x$$

Calculus 2 / BC

Chapter 8

Further Applications of Integration

Section 2

Area of a Surface of Revolution

Applications of Integration

Missouri State University

Oregon State University

Baylor University

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mhm. Uh huh. We got here. Oh, this is the one on the X axis. We have some function. Some function here fearful fix that is not necessarily positive. Us. We can see all day. The zero for why is here? So it is not. Yeah, necessarily. This is not always find out all the times. Always positive, right? Um, always Yeah, very non zero. So how we compute as to let me surface a surface of revolution by rotating? So that was a narrative is function from A to B. So while the function doesn't have to be, uh doesn't have to be positive for to obtain these surface of revolution by rotating destruction, because we can You can see here in the drawing. Just repeat, not to obtain something like this way don't need that. Are functions office positive? Because Paris, that surface there is, uh, well defined. So how can you give us a better? So this is the service off. We're evolution both, Uh, X. So you know, as you can see, saying that this point is c eso you help, but, uh, it will fix Well, if we fix our is this original function so you have that Fairfax is positive between mm on the sea. I'm careful. Thanks. He's, uh, negative between See, be so well, What we can do is we can You can compute this The area for this portion, which would be just the same area. Uh, if flags times, one less people. Okay. Fags squared. Yes. And then painting from a up to she to fight, I think for this one. Well, in just, uh, flip this part then to make, uh, this positive minus F effects is positive. So for this case, just put a minus on. Then these will be the area we've all seen. Thes number is the square. It doesn't matter. Just friends. Yeah, it's positive or negative. Yeah. Nobody is trying its positive or negative. The square is always Oh, great and honorable seal. So that this is what is fine. So we will do this part. Plus that part. Ah, inter from a TV off. That's all they is Sabbath for dysfunction. What this particular example is gonna be to fight central from a to B off the sea. Okay, folks. Bimes, Juan Pless. If frying squared the X plus two five, no themselves minus presenter from my area to be okay. One less if crime squared. Yes. So? Well, that works for this example. We have that that is becoming negative or a point. But if we have, we have, um, example off. Just the X axis is the white. I'm gonna have some points. A B. I mean, you have a better functions like these where these points are see on D. Well, by the same considerations, yes. So where there is the area operation about the X axis is gonna be the interval from a off to sea off if effects Bye. That was the square with one plus prime Squared X And then between C and D, we have toe flip the sign, affect certify, and it's minus f from see up to the tom off times one scored the one plus if prime squared the eggs just and then from here to be positive. So then So I've design from from the up to be so have the sign off is unchanged to be to buy times fix times the square root off front. Okay. Mhm. Yeah. Effects. Yes, On the wall In general, the same would happen if uh, the same happens. Um, before if they pass, uh, several intervals Where? Yeah, he's, uh, negative at several intervals. Several intervals. Yeah. So, as you can see what we're integrating here, this is the absolute value off effects on since surface positive on that segment, this is also equal to the absolute value off effects. This is also we go to the absolute body of the effects because along this segment, if it's positive for that segment positive along that segment, if it's negative, So in general, the more general way too right this interview is are just the internal off. The absolute value off affects on all this part remains and changes. So the absolute biological effects Times Square root off one plus f prime word X. And, uh so it doesn't matter if the function is all the time. Negative or just negative. Some intervals. This is the general form about four finding the area over on the surface of revolution. About the X axis about X axis General, for more, no matter what is the sign. Okay. What

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