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6 9) Sketch the region bounded by y Ex - x2 and the positive X-axis. (Hint for sketching thls region= note that y(X) intersects the x-axis at x and 1. To find the l...

Question

6 9) Sketch the region bounded by y Ex - x2 and the positive X-axis. (Hint for sketching thls region= note that y(X) intersects the x-axis at x and 1. To find the local maximum set and solve for x ]calculate the volume when this region is rotated about the line x = 2 b) Use the shell method Answer:Radius (2 - ) based on the diagram _ Note: dA (x - x2)dx: This gives us dV = 2r(2 x)( - x)dx:

6 9) Sketch the region bounded by y Ex - x2 and the positive X-axis. (Hint for sketching thls region= note that y(X) intersects the x-axis at x and 1. To find the local maximum set and solve for x ] calculate the volume when this region is rotated about the line x = 2 b) Use the shell method Answer: Radius (2 - ) based on the diagram _ Note: dA (x - x2)dx: This gives us dV = 2r(2 x)( - x)dx:



Answers

$9-14$ Use the method of cylindrical shells to find the volume of the
solid obtained by rotating the region bounded by the given curves
about the $x$ -axis. Sketch the region and a typical shell.
$$x=1+(y-2)^{2}, \quad x=2$$

In this area around the X axis. So X equals one plus y squared. Uh So that's a problem to that opens on the X axis. Um When X zero now X can be zero. When X is one, Y is zero. When X is too Why is plus or -1? Okay so we're going this way And x equals zero is the Y axis. And then white was one and white was too. Oh okay. So this is the area that we're talking about, we're gonna send it around the X. Axis. So you have to cut it this way for shells. So here's a shell. Alright, volume of a shell. Two pi R. H. T. Mhm. T. Is the thickness of the shell which is the the width of the slice which is D. Y. In this case? Okay. Or is the distance from the axis of rotation to the slice or from the axis of rotation to the edge of the shell? Which is also why? Or which is why? And then H. Is the height of the shell which is X. In this case and X is one plus Y squared. And then we're gonna pile the shells up from y equals one to y equals two. So the volume is two pi integral 1 to 2 Y times one plus Y squared. Do you? Why? That's two pi 1 to 2. Y plus Y cubed. Dy What? So that is Dubai Y squared over two plus Y to the fourth Over four. From 1 to 2? Two pi uh two squared over two thirds two plus two to the fourth over four. That's 4 -1 half plus 1/4. So two pi times 6 -3. Force, so two pi times 24 force minus 3/4 of 21 Force, So 21 pi over two.

Okay. What we want to do is walk through the process of being able to find the volume of the region bounded by UAE called the nine minus X squared X equal to zero. Why equal to zero about the line? Um X equal to three. Um And the very first thing we want to do is just go ahead. We should always, anytime you're finding area or volumes is too draw a quick sketch of that region. Um And so we know this is 369 So and we went only in the first quadrant. So there is our region right here that we are revolving about the line X equal to three. So I'm also going to draw in this line and we're evolving it about this line. X equal 23 Okay, so there is um some things and now the second thing we want to do is to go ahead and always draw in our representative rectangle and label key things. So here is our representative, right tangle. Um The with is delta X. The height of course is nine minus X squared. And then we want the distance of the center of the rectangle to the axis of rotation. So that is going to be um three minus X. So this right here is going to be three minus X. Okay, so then that volume is equal to two pi that distance from the axis of rotation to the center of that representative right tangle times to hide through the rectangle times the width of the white tangle. Okay now let's go ahead and multiply out these both of these Bino meals. Um So this would be to pi um this will be 27 mm. I don't think that's going to be nice minus nine X minus three X squared plus X. Cubed delta X. Okay, so now my volume, it's going to be actually equal to the integral of that. So my volume is going to be the integral and we're going from 0 to 3 because that's the region. And the integral of bring out this two pi here um of 27 minus nine x minus three X squared plus X cubed TX. Ok so now we know how to integrate all of that. Right? So this is going to be equal to two pi. This is going to be 27 x minus nine. Have X squared minus X cubed plus I believe 1/4 X. And we evaluate all of that at three and then subtract it, evaluated zero, which you notice that I evaluate all of this at zero. That's just gonna be a zero. So the volume is going to be equal to two pi and then this is going to be 81 minus 80 ones over to minus 27 plus 81/4. And I think this kind of reduces down to 1 35 pi over to and there is my volume.

Okay. What we want to do is walk through the process of being able to find the volume of the region banded by Weichel the nine minus X squared. Um For X values greater than or equal to zero, X equals zero. Y equals zero about the Y axis. And any time we were talking about finding areas or volumes, it's always a good idea to draw the graph of the region. And so we're going to go ahead and draw that graph, it doesn't have to be perfect. Um We know it's going to be up here at nine. So if this is 369 and then it's going to cost the X axis that positive Three. So here is that region right here um without three and that is nine and we're revolving it about the Y axis. Um And so what we want to do is so it's going to be this region right here. X equals zero. Why called zero? And the graph um then the next thing we want to do is always draw that representative red rectangle. Right? So we have our little representative rectangle right here. Whoops. Uh huh. I can't draw a straight line for the life of me. Um So there we go. Um where this with is delta X. The height is of course nine minus X squared. And the distance from the axis of rotation is X. So um there is are a kind of a good quick sketch. So delta V is equal to two pi two pi X because that's the distance from or X is the distance from the axis of rotation. Um And then we are doing the height which is nine minus X squared. And then of course delta X. The width of that rectangle. So the volume is the integral from 0 to 3 of this. Let's go ahead and let me go ahead and distribute the experts. So this is two pi nine x minus execute. So this is going to be two pi times nine x minus x cubed and of course the delta X goes into dx OK, so now we can go ahead and integrate some a factor out that two pi and then this is going to be nine halfs X squared minus 1/4 X to the fourth and we're going to evaluate both of those at three and zero. Um So this is going to be two pi this will be 81/2 minus 81 over four. I'm in of course minus minus zero. Right, so everything is over zero. Um and so then this will be actually equal to, this is actually gonna be equal to um 81 pi over two and there is our volume.

It's in the area enclosed by this curb and X equals zero about the X. Axis. So we have to do some more caramel effect out the white squares or get four -Y. So when X. Zero, why is zero or four? Um When X. Is let's see when y is too then X. Is eight And when Y is one X is 1 -3 three and the next is three. Why is nine? Um That's why I say maybe something like that. All right. I want even if that's not exactly right. It's good enough. It gives us the information we need. Okay that's the shell. The volume is two pi. R. H. T. T. Is the thickness which is the thickness of the slice or the thickness of the show Which is the wife. R. Is the distance from the axis of rotation to the slice or to the edge of the shell. Which is why h. Is the height of the shell which is X. Which is for Y squared minus two Y. Cute. And then we're going to stick them up from here like zero. So y equals four volume equals two pi 0 to 4. Why times four Y squared minus Y cubed the wife two pi 0-4 for y cubed minus Y to the fourth. B. Y. Two pi For Y to the 4th over four. Why is why to the 5/5? From 0 to 4? Uh huh. So I'm going to put two pi Y to the fourth Time is 1-. Why were five from 04 254 to the fourth times one minus 4/5 two Pi 4 to the 4th times one. Fifth Forward to the fourth is four times for 16, times two, so I get 5 12/5 pi.


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