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Find the volume of the solid obtained by rotating about the y-axis the region bounded by y = 3r y = 0.and(a) Atypical shell has radius circumference 2Tx; and height...

Question

Find the volume of the solid obtained by rotating about the y-axis the region bounded by y = 3r y = 0.and(a) Atypical shell has radius circumference 2Tx; and height f(z) Preview(6)by the shell method; the volume isV =dxPreviewand(c) Evaluate the integral: VPreview

Find the volume of the solid obtained by rotating about the y-axis the region bounded by y = 3r y = 0. and (a) Atypical shell has radius circumference 2Tx; and height f(z) Preview (6) by the shell method; the volume is V = dx Preview and (c) Evaluate the integral: V Preview



Answers

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.
$$y=x, y=\sqrt{x} ; \quad \text { about } y=1$$

A base of discretion. We have to find a whole human for a soul of revolution between the curves Why is equal to one minus X squared Y is equal to zero on the X axis. So if you were to draw Big Sketch So this is the line. Why is equal to zero by zero, which is the X axis and this is one minus X squared. This is one minus X squared. So we have to basically imagine this revolving around the X axis and invisibly forming a sphere. So the volume for soldier of revolution is the integral between X is equal to negative one on one over one minus X squared DX. This is equal to X minus X huge over three. I'm gonna evaluate this from naked. I wantto one, This is one minus 1/3. That's one minus 1/3 minus negative one minus negative, one over three months, plus one over three. So therefore, our final and Zaheer, don't forget, we also multiplied by pi. There's a pile on the outside. So our final answer here is equal to 16 over 15 over 15 multiplied by pi

So in this problem, we're finding the volume of the solid obtained by rotating the region founded by a Y equals execute. Wyffels X with X is greater than zero. So we can first start off by drawing on the reading. I'm going first. You'll notice that I put the majority of the area in the excess positive. Gary and Wyatt was positive area. And the reason I did this is because we know that excess positive and when X is positive. Why is also positive? Because pies with Cuba is always positive, as is a pilot raised two to the power. So the majority of the action will be going fun going to happen in the first clock ins. We can first draw the y equals X line that was like that. And if you think about it, y equals acts y equals execute those two flying slash curves intersect at the point at one one. They also intersect at the point zero zero because both both of those points satisfy the equations of the line. Zero Q zero and also zero equals zero. Once cured, this one in one equals one. So where you draw out the second line white because ex cued, this point will be one one, and this point will be zero zero. And we're rotating this about the X axis. So let me first off by drying start by highlighting the region that we're rotating and red. We're rotating this about the X axis, which means that we are forming a washer, which is made out of subtracting the smaller disk out of a larger disc. So here is the radius. So as you can imagine, there's a smaller disc. Here's the larger disk. We only care about this region between the disks. Thus the reason why we call it a washer. So to do this when you know once again find the cross sectional area on in this time because we're rotating about the X axis when you find it in terms of X, because as exchanges, the area changes as well, because the washer this time it's a circle subtracted from another circle. So this is equal to pi r squared and you can make the bigger circle of big R. The smaller ship will smile, So as you can see here, the y equals X line, which is the top line right here is creating the bigger circle. And then the Y equals X to the Q X to the three line or a curve is creating the smaller circle because it is less than the Y equals X line below it, pretty much the entire in a roll from zero to one, and that makes sense when you keep a fraction, you make it smaller. So the are right here represents the y equals excellence We can put that in. So our is just why, in this case, a just plugging X and then the small R is just why equals excuse plug in skewed and that is equal to pi. I'm pulling out the pie here. So the X squared minus X to the six. So that is our cross sectional area function in terms of X, so we can go ahead and do our anymore. Now, as I said before, we're integrating from X equals zero two X equals one and we put in the function that we just found. So I'm going to immediately pull up. I hear because pie has no relation with the ex, Then you can put an X squared minus x for the six d x because that is what we found before to evaluate this interval when you find the anti derivative of X squared in excess of six. Oh, that is just X cubed over three minus X to the seven first seven Valentin from zero to one. We don't really care about the zero case because that was his Give us zeros. So we only care about the one case. And when we do that, we get pie. I went there minus one sex. And if you do a little bit more math, you'll finally arrive at the answer, which is four pi over twenty one.

Were given curves and line and ratifying the volume of the solid obtained by rotating the region down by these curves. About this line were asked to sketch the region the solid and a typical disk or washer the curbs are Y equals six minus x squared Y equals two. And the line that we're rotating is about the X axis right? Yeah. Their rules on. So first I'll sketch the graph. So this first curve is a parabola. It's downward facing. It has a a vertex at 06. Who else is? Mhm. Then we have a horizontal line. Y equals two located about here. So the region that were interested is this region here in red. I remember you we want to find these endpoints here. Well we know that the why coordinate for these endpoints is too. So if you plug in to to our parabola we get two equals six minus X squared. So that x squared equals four and therefore X equals plus or minus two. So these points are negative +22 and positive to to now the X axis. Is this horizontal line here and therefore if we rotate about the X axis, we get a shape that looks something like this. Flash. You get home started, right. Try to go overseas. Like it was from her brother. I never got home started. Like it was all Flash was Flash, but I never thought it was funny. And I also thought it was like, isn't that like Cruz announcer? Yeah. Like I thought homes are was just supposed to be someone with down syndrome and your father. I mean I thought it was supposed to be literally something went down advanced. Said that some. Yeah, they would just like change like words word. It's a good job. Mhm. So this is sort of what the solid looks like. And of course if we label that's a lot of that. So a washer in red, we're in green that student green, it will look something like this. Didn't get to get a computer that is all you had. Yeah. Well we don't get a microwave and learn how to use. So this green part is our washer that's the best suited. And therefore jiffy pop our washer we see has in inner radius Of two and an outer radius. Which is, you know the other thing I know we that's a function of X. It's six minus X squared. Like so our volume is the integral using the washer method pi times the integral from X equals negative 22 X equals positive too of the outer radius squared. So six minus X squared squared minus Dina radius two squared D. X. And if you multiply this out and simplify, you eventually get after skipping a few steps. I'm not gonna do it here. This is 384 pi last season, the last season, so last episode over five

Okay, So this question wants us to calculate the volume of the shape that's formed by rotating the area under the graph of f of X equal see between zero and are around the y axis. So part eh wants us to do this without integration. So we could just think about this conceptually. So for evolving this region around the y axis, we're taking slivers of this graph and spinning them around a form cylinders. But in this case, since the function is constant, we're just going to get all the slivers at the same height. So it's all the shelves at the same high. So it's just going to form one big cylinder of height. See, and radius are so part, eh? The solid is a cylinder of radius are and the height equal to see. So recall that the formula for the volume of a cylinder, his pie r squared eight. So in this case, R equals R. And our high is equal. See, So volume equals pi r squared psi. No, it wants us to find this using integration show method. So using the shell method, we need a radius and I so our radius is simply the distance from the origin to the graph, which is X so radius we'll use a capital R X, and then the height is just the height of the graph. Each sliver is Justus tall as the graph of like will see h equal see. So then volume equals two pi times the integral from zero to our of radius times, height, times, DX, which equals to pie times see X squared over two from zero to h Sorry, zero r and the twos cancel. So this just gives us pi r squared psi, which is exactly what we got in the previous part. And you'll find that a lot of formulas air like this. You could drive them geometrically or you can do them or rigorously with integration.


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