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Summer Assignment 3: Problem 15PreviousProblem ListNextpoint)Using disks or washers, find the volume of the solid obtained by rotating the region bounded by the cur...

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Summer Assignment 3: Problem 15PreviousProblem ListNextpoint)Using disks or washers, find the volume of the solid obtained by rotating the region bounded by the curves y = sec(x). y 1*=-.and x = about the X-axisVolumePreview My AnswersSubmit AnswersYou have attempted this problem 0 times_ You have 3 attempts remaining:

Summer Assignment 3: Problem 15 Previous Problem List Next point) Using disks or washers, find the volume of the solid obtained by rotating the region bounded by the curves y = sec(x). y 1*=-.and x = about the X-axis Volume Preview My Answers Submit Answers You have attempted this problem 0 times_ You have 3 attempts remaining:



Answers

$1 - 18$ 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.

$$
y = x ^ { 3 } , y = x , x \geqslant 0 ; \quad \text { about the } x
$$

We want to find a volume generated by this. Why equals to sine X from X equals to 02 X equals to power tree revolving around the excesses. Now, if we were to sketch this region here from zero to pi over three. Okay. Of the second ground you can see that it would be something like this. This is why equals two second picks. Now if I were to take a small strip over here, the week is so small. Our called D. X. Or delta X. Now it's why value will just be why? As it revolves around the excesses, it will form a disk. A very tiny cylinder of the high the X over here. And the radius is why? So the volume for this cylinder? Little cylinder over here will be hi the radius is why square. And the X. Now I want to add up the volume of all this little disk from X equals zero to poetry. But because the beef of this, this is so small, it will be a continuous submission which is the integration sign from zero to pi over tree. So this is the volume over the excesses. So now we want to find the volume. It will be zero two pi over three. Why square the X. With the pie in front? So we'll be pine zero poetry. Second square ex D. S. Now we know that when we integrate second square we will have bend your necks. And so subbing in the upper limit where X. Is poetry, you have tension poetry minus. When we start in the lower limit on zero into the X minus tension zero. So this will be pie. Yeah rotary minus zero. So our final answer is route tree pie, unique. Cute. Yeah.

Were given a set of curves in the line and were asked to find the volume of the solid obtained by rotating the region. Bounded by these curves about this line curves or why equals X Cubed y equals X and X is greater than or equal to zero. The line is about the X axis first straw, the region in the X Y plane that will be rotating. So this is going to be in the first quadrant. So why equals X cubed as points at 00 11? Michael's ex also has points there, so really focusing just on this one small region. Where is y equals? X Cute has a value of 1/8 at one half. Why equals X is the value of one half one half. And so this region in red is the region that we're going to be rotating about the X axis. In order to do this, I'll draw a second graf, which contains the 1st and 4th quadrants. So we get shape that looks something like this. So this is the solid. And if we take a cross section of this solid, we see that it's in fact going to be a washer. It looks something like this. So looking at graph of our solid a cross section is a washer or analysts, inner radius X cubed and an outer radius of X. Therefore, the area of the washer, The FX is pi times X squared minus pi times x cubed squared which is the same His pi times X squared minus six to the sixth in the volume of this solid is the integral from X equals zero to x equals one of the area DX, which is the integral from 01 of pi times X squared minus X to the sixth, the X taking into derivatives This is pi Times one third x cubed minus 1/7 extra seventh from 01 Substituting This is pie times one third minus 17 which simplifies 24 20 firsts times pi

Were given a set of curves on the line and were asked to find the volume of the solid obtained. I rotating the region bounded by these curves about this line the curves air Why equals execute bye equals zero and X equals one in the line were rotating is about the line. Why he was X equals two Sorry. First, let's sketch a graph of the region edited by these curves. This is mostly in the first quadrant, so we have a line X equals one and vertical line and X equals two. Now we have the function like was execute disappointed 00 and at 11 and it has a shape like this. So region that we want to rotate Is this region in red? Here in the line of rotating around is this dashed red line. So the sketch, the solid I'm going to just need the first quadrant. I'm gonna change the scale a little bit for clarity. So this is what the's solid essentially looks like. And this is what a cross section of the solid looks like. Looking at the cross section, we see that it is a washer with an inner radius which is the top function. X equals three minus the bottom function X equals to which is one and has an outer radius, which is top function to minus the bottom function, which in this case could be solved for X equals the cube root of y and therefore the area of this washer is pi times tu minus the cube root of y squared minus pi times one squared This simplifies two pi times four minus four Q Bert Y plus the cube root of y squared minus one, which is the same as pi times three minus four cube root of y, plus the cube root of y squared. And therefore the volume of this solid is the integral from y equals zero toe, one of the area of the washer, integral from 01 of pi times three minus four times the cube root of why, plus the cube root of y squared de y. Taking anti derivatives, we get pi times three y minus three y to the four thirds plus 2/5. Why to the five halves? Sorry should be 3/5 y to the five thirds from 0 to 1. In substituting, we get pi times three minus three plus 3/5 which simplifies to 3/5 pie

This area about the X axis and then find the volume. So y equals the square root of x minus one explains one greater than or equal zero, X greater than or equal to one is its domain. So I started at one when X is one Y zero Where next is too wise one. When X is five, why is too, so that's how I got the graph And then y equals zero And x equals five. All right, so we're going to have to cut vertically. Yeah, volume is pi r squared, H. H is the height. Oh here's here's one of the disks done evil in it because um because the area completely touches the axis of rotation, H is the height which is this or the thickness of the slice? Which is dx cars from the axis of rotation out to the edge of the disk. So you can see that. It is why on the top -1. On the bottom one The bottom of zero. So why on the top which is squared of X -1. Then we're going to pile them up from x equals one. two x equals five. The volume is 1-5 squared of x minus one squared Dx hopes up throughout the pie. Hi r squared H. So that's pi integral 1-5 x -1 D. X. That's pi export over 2 -1 From 1- five. That's pie 25/2 -5 -1/2 -1. Pie. 25 has -1 half. That's 24/2. That's 12 -5 -1. That's -4, So I get 85 for this one.


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