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Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Then use your calculator to evaluate the integral correct to five decimal places.

$ x^2 + 4y^2 = 4 $

(b) About $ y = 2 $

(b) About $ x = 2 $

a) 8$\pi^{2} \approx 78.9568$

b) 8$\pi^{2} \approx 78.9568$

Applications of Integration

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Baylor University

University of Michigan - Ann Arbor

Idaho State University

Okay, first I need a picture of this thing and if it didn't have this for in front of the Y squared, it would be a circle whose radius is two and center 00. But with the four there, I know it's any lips instead. So it's X squared over four plus Y squared over one equals one. So when I draw it it's too on the X axis, one on the Y axis and then it's daddy lips. Okay, so first we're going to send it around the line, Y equals two, which doesn't touch it. So it's going to go around and will make a kind of a ring looking thing, like a like a baby floaty, maybe like the kind they wear on their arm which is kind of like that. Okay, so how should we cut it? Well, if we want shells, we're gonna cut it horizontally. If we want disks, we're gonna cut it vertically. I'm just gonna do disks. Okay, so I'm gonna cut it like this. So I send this piece around and it makes a flat washer and it's real big because of the distance here. Okay, like this. And because this right here isn't touching this dotted line, you can see that's going to make a hole in it right there. Oh okay. Okay, let me give that whole another try there. Okay, that's a little bit better. Okay, the volume of that. Think washer is pie big R squared minus little R squared H. And H. Is the thickness of this piece right here, which is the thickness right here. What you can see is D X. Okay. And then are is the distance from the center to the outside of the washer? All right, so it's the distance from here to hear them. All right, so we know that from here to here is to and then from the X axis to to the outside here is some Y value. So big are is two plus why? And then little R is the distance from the center to that um edge of the hole which is right here. So we know that that one is also too but this time it's minus that Y value. Okay, because it's the difference between those two or this green arrow plus this Blues quickly add up to this Blues quickly. So little Rs two minus y. Okay, so we can't put that in because we're going to integrate with respect to X. So we have to get the y out of here. So I'm going to solve the equation for why? So I get y squared equals one minus X squared over four. And so why is plus minus the square root to one minus X squared over four? But the way I haven't written here, I'm going to use the plus both times I'm going to do two plus y and then to minus y. So big are is two plus that square root And little r is 2- that square root. All right, so then it's magically gone. Okay, there it is. Okay, so then the volume is going to be the integral pie times. Big R squared two plus one minus X squared over four quantity squared. Uh huh minus little R squared which is two plus the scoops. Two minus the square to one minus X squared over four squared D. X. And then we're stacking these little little uh disks washer things up from X equals negative 22 X equals two. So we could do -222 or we could do twice 0-2. So let's just do that. Make it easier. So it's two pi 0 to 2. Let's go ahead and simplify a little bit and then it won't be so hard to type in the calculator. This will be four plus four times that square root plus that that square root squared minus. You gotta be careful here for minus for that square root, one minus X squared of four plus one minus X squared over four. And then dx at the end oh two pi 0- two. Okay this four and this four cancel because it's 4 -4. This 1- escort over four that cancels with that. So we have four minus minus four. So we have eight square root of one minus X squared over four D. X. Or 16 pie. 0 to 2. One minus X squared over four Dx. All right. I don't have the kind of calculator where you integrate? Oh I guess I could have could have found an integral calculator on the on the computer but find something that will integrate that and then out will come the answer. Okay now let's send it around the line x equals two. Okay so remember it looked like this so now we're going to send it around this line um shells or disks. Let's just do disk again. Since since we know how to do that. Ah No let's do let's do shells. That will be more fun anyway. Oops. Alright so to make to make shells I'm gonna cut it this way. Okay and then I'm gonna send it around. Okay so here's one of the shells and then there's a little tiny one here and then okay this looks sort of like a a big fat doughnut that is so puffy. The whole got completely filled up right here. Okay. Okay. Volume of shell two pi R. H. T. Okay so here's the shell T. Is the thickness and you can see the thickness here is D. X. H. Is the height from the bottom to the top. So you can see if I draw one of these. The height is why? Plus why? So to y. And then r. Is the radius of the shell? So our is from the axis of rotation out to here so the axis of rotation remember is to and then this is some X. Value So that radius is two -X. All right and then remember we saw for why? And we got plus or minus the square root to one minus X squared over four. Okay so that's what's gonna go here. All right so the volume is two pi integral are H. T. And we're stacking these shells up from here all the way to hear. So -222 again. Okay so we can do zero this time. We can't okay we can't do 0-2 and then do it twice because this little shell is way different than its partner out here. It would be really big like that. So you're gonna have to do -222 on this one. Uh Simplification you could do. Oh I'm a domestic right there. Oops mm I can remember that X only the X squared is under the four not the one But you could do that to Pi -2-2 to -X. You could write that as four minus X squared over the square root of four. So this too and that would cancel. All right so now that's ready to go. So put that in your calculator and see what you get there. Okay I hope that health and didn't confuse you with this um the way that I did the volume but it's it's really easy to think about this way if you just always remember one is two pi R H. T. And the other one's pi r squared HH

Oklahoma State University

Applications of Integration