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.
$ y = x^3 $ , $ y = 1 $ , $ x = 2 $ ; about $ y = -3 $
Applications of Integration
For this question, we are finding the volume of a solid rotate, uh, obtained by rotating the region bounded by y equals X Cubed y equals one and X equals two. We can first begin by drawing out the axes now weakens. First, start off with the Y equals one line. That's pretty simple. It's just a horizontal line at one. The exit was to one, is also pretty simple. That's just a vertical line equals X equals two. So this is kind of out of scale. Is this one? This point is to this is wine. So finally we could do the Y equals execute line. We know that one one is a point zero zero is a point, and also here native. One native won his point and to eight is a point. So it's all the way up here, so you draw very steep crapola. It's not really a problem. It's have a problem because the other side the flexed outwards because the Negative cube is always negative. Now you know the region. This part covered it in red. We need aligned to rotate about, which is why equals negative three. Let's just call this point. I know it's not to scale, but I'm gonna fit in the screen. This will be ninety three, but we can also this point call this point on. There's eight. So we know that this point is one one that is two eight and that is two one. So we know that three points of intersection that creates the region. So we're rotating about the y equals negative three line. It's cream washers. This is the radius of the washer. So when you come up with the cross sectional area and as you can see, it changes with respect to X or hear, the radius is near zero. Over here, the radio is much larger. So let's come up with before you changes of the District X, it's the equal to the larger area, minus the smaller area, or the area of this larger disc. Finest area of the smarter disc. Pull out. Uh, hi. Now, if you see here the smaller disc, we're gonna first began. Minor dissing is always is always constant because while exchanges the distance between the Y equals one line and the Y equals native Syrian, wine is always the same. And that's four. So we know that little R is always for we'LL write that to decide. No, the curve that creates the bigger disc is this cover here, and that's the white goes Excuse mine. Now the distance from the y equals X Q line. We're not lying. Curve to the y equals negative. Three Line is whatever the Y values, this is at any point. Plus three. Virginia at the distance from the Michael zero to the Y equals negative three line, and that's always three. So that's just execute plus three, that's what. Big Ari. So we didn't talk about him. Now, if you must buy all this out, you get the phone about the pie. Still beginning X X cube squared is exit six three times X cubed times two. There's six ex Hume, and then three square is just nine four. Square is sixteen equals pi x two, six six. Execute minus seven. And so that is our cross sectional area, which we can put into our interval, the stacking them all up in getting the vine. Now we've already figured out the points of intersection, so we're integrating from X equals one two and people's, too. I want to immediately pull out the part because it's still constant. So I'm gonna put in the cross sectional area that we found. So we need a value with this interval, which is relatively simple, keep the pile in the beginning and evaluate anti derivatives. Anti drug Exit six is extra seventh over seven, the anti derivative of X to the three. That's point right there. It's just X for before you still multiplied by six. So you get three over too excellent fourth and and centre of seventy seven x, and this is all still evaluated at one and two. There's two right there. No wait. It spread. First plug in X equals two. Gino. The seventh is one twenty eight pound and then six were rather three times, too, to the fourth divide by two. So that's really just three times two to the eight two to the third. Because the two camps, the Divide by two, cancels out with one of the twos of two to the fourth, so become three. So it's three times two, two, three, two three eight. So three times eight is twenty four minus seven times two is fourteen. Knowing you evaluate when x equals one that's just one seven plus tree house minus. And when you do with this all out, your final answer is or see anyone Bye over fourteen, and that's your final answer.