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Show that if we rotate the curve $ y = e^{\frac{x}{2}} + e^{\frac{-x}{2}} $ about the x-axis, the area of the resulting surface is the same value as the enclosed volume for any interval $ a \le x \le b $.

$$V=\pi \int_{a}^{b} y^{2} d x$$

Applications of Integration

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Missouri State University

Campbell University

Harvey Mudd College

University of Michigan - Ann Arbor

of this curve. This function, that is why is he going to eat there? Ex cops must minus X hops if you draw the X Y plane X y axis. But there it sounds, Kirk, something like these See magic. So you want to consider We have points a me We rotate this region about the x axis So you're datable bx We're gonna obtain shape That looks like these, Like, make some sort of, uh, tears come like that. Todo Hakkies This vision you want to compute? What is the area? What is the area? Yeah, which in the end way would like to see that is equal to the volume enclosed, maybe Syria. So because this this area of t the volume enclosed by deep division so that we don't get this area is the phone. So you have these we call this, we can call these effects. So this area would be, um oh, people toe to buy from Central from pay off to be off. Careful fix very soon is part of DS. That would be one plus f prime squared. Ah, the X So for dysfunction or crime is gonna be ableto Yeah, that that would be accepts. Thanks. We've been making your video. Uh huh. 41. You have also more to the minus six house. But by the negative way, Get up. Minus off. So at this object here, this square with the one plus, If Frank is gonna be called to square one plastic right square, be equal to the square. Root off one plus, Mrs Square, which would be neat to the X. Uh huh. Minus, because abstract, we have common denominator minus three minus six. House. Well, that is squared, divided by the square. I'm sort of this sequel to square root off one plus, we'll this is Square is gonna be this Times Square, which you see today three x x squared shops squared C three X. Uh, the middle time is the next time is two times the product of these two, which is one so minus two on the plus that square. Plus, you didn't minus X and then all of that very by four. So that we are these two. These two guys, we have the square root off before, plus you three x minus two plus e to the minus X on All god divided by four so that we can simplify that today. All these four on that my students can to cancel. And then you have this time today Thanks. Not last but implored my school plus if u minus X and law, it's going before it is too. Yeah, to have that then, uh, so the area would be able to and yeah, to be able to five. I'm sitting to go from zero from my from Malone Toby off our function, but he's beat the eggs hops plus into the minus six house That is our function. No there And then that attempts. This is corporate. It's great little between x Last two he didn't minus X and then that got about two the X. So one thing that we can do is we can cancel these couple off twos. Well, we can note that inside we have this time conditions The square that is able to do this, Yeah, if you squared up without pain is Times Square, which you see to the X, the middle time to time. So the middle time is this times that which is 12 times. So does that mean this term square is uh, Ethan, minus six. So that with the square. Well, uh, cancel because the term is positive. So this area is gonna be able to fight. Found some the interval from They have to be off you today. X squared. Plus needed in minus X squared. But also about 10, then from the square. Because since we have this quit, square it off a square is the term itself. So it would be that term eggs into the X hops. Plus, you didn't minus X hops? Yes. Then we can see that this term is the same. So this term is equal toe. You need the eggs, hops los E to the minus next house square. So what? Well, if he rotate this area, we can see there. Uh, what would be the problem in that region? So you can am be so we'll have some. Yeah, it's important. Like these Unless you are a in the are the same. If you rotate that vision, you have something like these shape the same. I said well, but if we were now to consider what is the volume off these visions of? We need to our each of these vices here it's okay. Transfers all cats here Business license and one slides off this for a particular X here without area five EMS or the radius squared. But the radius is precisely thesis The radius is our function to be bye Them star function squared which is the same interval that we have here now here by Inter from may have to be careful Fix squared the X which is well to complete the volume, we would need just toe all those Fine. So for so these two are equal on the reason why Daddy's while those two are equal because for that particular function one that that into sequel toe thes interval there waas the regional interval to compute the surface area. Um, like this. Yeah. So for this function, it happens of these two intervals are the same for any values of A and B which was Well, this is the surface area surface area about rotation. The surface area a rotation of example is equal to the volume. Yeah, off also reporting the graph below X. So it is because off this with a particular function for apple effects equals 2 ft Ex house not seeking a minus six cops. Yeah, yeah,

University of Colorado at Boulder

Applications of Integration