Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.

$ xy = 1 $ , $ y = 0 $ , $ x = 1 $ , $ x = 2 $ ; about $ x = -1 $

$(2+2 \ln (2)) \pi \approx 10.638$

Applications of Integration

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Oregon State University

Harvey Mudd College

Idaho State University

Boston College

we're funded. Volume of solid, obtained by rotating the region, bounded by X y equals one y equals zero X equals one and X equals two. You start by drawing off the region. The X Y equals one is pretty simple. We know that these two points are in the curve. Won one in native one negative one as exit parties and fanny, Why push zero X approaches zero my purchases. Positive infinity. It's the same on the other side. We don't really care about that because the region doesn't really on Doesn't really deal with the negatives. Now the X equals one is just a vertical line. There it was too, is not viable line right there. X equals need one river line right there and the Weikel Zeros X axis. So I'm not going to draw it out. We get is a region with thes for intersection points these four intersection points or one one because X Y equals one, um, one one satisfied that along with X equals one also satisfies that this point is to one half cause that satisfies both. The X equals X Y equals one and the X equals two qualification. This is two zero X is one zero on those air. Pretty simple, rotating about this way. So we get washers. Now when you look closer thiss, um this particular region and it's at its shape. What we get is this shape over here, and we're funding how the radius our changes as you change Why? And you know oratorio do that. Ah, you know, is that this region is actually two regions. You call this region one call this week too reaching one. It's pretty interesting, the only integral to solve it. Reason, too. It's just a rectangle. But I'll still used intervals to solve it because this is an integration exercise. Nevertheless, we can start with region too, and find the cross sectional area through the bigger area minus the small area of the disks. Now the area is always constant. In particular is the distance from here to here. And that is the distance from the X equals two to the X equals negative one mind that is always three. Now the distance of the smaller radius. It also same and constant in region two only though, um or in particular, because that's what we're looking at. The distances from here to here or the X equals one line to the X equals negative one mind. There's always two. You know this is nine. This is four. We get nine pine minus four pi. That's five pie. Now we can look at each one. Find the cross sectional area. It's a little more complicated for me. It's still the same speaker disc, one smaller disc, but the bigger disc area radius is different now. If you look here, this car is creating the bigger disc. So we express, um, this curve in terms of the X coordinates, because that's what we used to find the distances between on the vertical lines. It is pretty simple. You divide by both sides by why you got X equals one over. Why no finding the exporting of this curve gives you the distance of any point on the curve, too. The Y axis. But that's not what we need. We need a point on the curve to X equals negative. What line? So we have to add one toe, whatever the exporting it wass. So we add one to take into account. The X equals negative one distance to the Y axis smaller are Well, look at this region Still the same. It's just still too. Now we couldn't solve Distribute whenever I square plus two over. Why this one? Minus four pi We'LL fight This is pie over one square You wonder why history So that is the cross sectional area for Region one. Now we could do our universe the star ofthe region to again If you look in this region Um when you find three specific points here, here and here the y values of those because that's what we're integrated over. If you look at this overall, we'Ll soon find out that the bottom is just zero midway point is one half and that's how point is just one for the y values. So for this, we're going from zero to one half and put in the cross sectional area that we found here himself. Look, the pie. You're in one for five years. Really? We could just what? Five itself to get you. Why, it's five pie. Why value added that one happens hero. That's just five pie to zero This five over two. It was a simple interval. Let's do the harder in a row. This time is going from one half to one for reading one. Well, at the pie from the very beginning. If you look at this when you plug this in right here, let's put that in. You find the anti Drew this integrative off this right here. Now that's just another way of expressing wide, maybe two. So now we have two plus one. Isn't one divide by when you divide by Meghan When you just get anyways So maybe one of her why plus to a land? Why has the inspector of two times whenever Why my story? Why this is it away. One half and one. Now we need first plugin When y equals one, it's gonna be pretty long. Let's give ourselves a lot of space when y equals one native one over one one Two times, Ellen one and mystery over that mines when y equals one half native one negative two again to Ellen, one half my tree house. So when your valley on this and simplified, you can keep the pile on the outside. Um, Ellen of one is just zero, so we can get that, uh, native one minus three is I need it for, um, we're adding to here is attracting two and one half and adding great Hera, the house that is native to minus to Ellen. One half, um, we can simplify. The Ellen is what happened. Because there is this rule where Ellen of a over B is equal to Ellen a minus. Eleanor. So Elena won't have is equal to Ellen of one. You know what that is? That's just euro. I don't have to. So we really just get equals Negative Elena, too. So, Eleanor, one half is equal to negative. Eleanor, too. It's just we're adding two times. Balan off to clustering over to. Now, let's come up with a comment in on there. So but there's all under over too. This now becomes a four. This now becomes a negative. What? So this becomes equals five times. Four plus three is negative. One plus for Alan two. Oh, over to you. So that's for Region one. We still have to go back and add this. So let's go. Five power for two. Plus, they're who fund here. She's nice. High times, um, native one plus four. Ellen too. Two. What we get is part the pie five minus one was for Ellen too. All over, too. And matches for. And you could divide by two. So all simplified the pie two plus two, two. And that's your final answer. If you wantto, uh, calculated to the decimal place, this is equal to ten point six three eight. Both of these answers are acceptable.