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Unc nEton bounded JW {+.> the volumne ofte solid KCTleruted nocif Wciton TOLilu_WAI [rom W TCElual Atohetura AeeteneolurdFind the volutte of the solid Fencraled ...

Question

Unc nEton bounded JW {+.> the volumne ofte solid KCTleruted nocif Wciton TOLilu_WAI [rom W TCElual Atohetura AeeteneolurdFind the volutte of the solid Fencraled Yuls muling tne nelon Skelch the solid ud specify the mcthv/ YousceJelneeEnolneFind the volume of the solid generated by rolating the region A 4> delined (5a) above pbouuthalia Y= Skctch the solid and speeily the method You usedvolume generated by rotating the region _ 15 delined above = (7d) Find the Specily the method YOu usedJa

Unc nEton bounded JW {+.> the volumne ofte solid KCTleruted nocif Wciton TOLilu_ WAI [rom W TCElual Atohetura Aeeten eolurd Find the volutte of the solid Fencraled Yuls muling tne nelon Skelch the solid ud specify the mcthv/ Yousce Jelnee Enolne Find the volume of the solid generated by rolating the region A 4> delined (5a) above pbouuthalia Y= Skctch the solid and speeily the method You used volume generated by rotating the region _ 15 delined above = (7d) Find the Specily the method YOu used Jahouutt lingX EeL



Answers

Find the volume of the solid generated when the region enclosed by $y=\sqrt{x+1}, y=\sqrt{2 x},$ and $y=0$ is revolved about the $x$ -axis. [Hint: Split the solid into two parts.]

We have this washer method and shall method the Harding toe. This we have their tradition accesses the X annoy from this washer and shall method the X and y from the game. And you have on I could shown off integrate clothes. My wife calls Yeah, statistical to skirt off X plus one. We have also given why it calls Square it off two x and why call sero? So we have revolved at X axes on a revolution and then you have points We're in the intersection off just toe equation So we have great both sides of the equation Square root off X plus one we call skirt top two x squared both sides of the equation So we have export So on that the Sequels toe X simplifying the X equals one substitute one of those situations we have. Why that is equal spirit off one placement goes to So why constant? So we have intersection at one square it up to for this crap off Mikel squared up experts on Michael's two X and zero Michael zero This is the intersection you need toe volume liberalism that on my exactly So so the sum so how? Use this word. Use that formula. This one So that Russian methods you have equal spy, right? For, um, we get the values from from X houses. This one from 0 to 1. Because the X is here. Yeah, 0 to 1. So you have using this formula here. So square it off. Experts one square that it's supposed to express one. We scared it. Right? So cancel that. And then you have my nose. Five time from 0 to 1. Scott, Reducto X squared. So cancel. Cancel. So we have two X. This is the X and the X. So from this we have expressed one square Alberto, my nose toe expert The work so that also it calls toe expert. We saw us the group the pie. So from 0 to 1. So we canceled zero here. So one plus one equals 22 squared calls. Poor portable, but two equals two minus once critical is one minus zero. This is spy. So the result is volume, but physicals toe by cubic unit. So the volume is this

All right in this problem, we wish to sketch the region are bounded by the given curves and utilize the shell method to find the volume of the solid revolution generated by revolving are about the Y axis where the corresponding our region, our wives critical X. X equals Y. Plus. To this question is challenging understanding of applications integration namely the volume of solid shell method about the Y axis gives the equals two pi Interrelated B x F minus G. Where athletes greater than G on A to B. The determined A B, F and G. We first grab our our is given here as the region in red and yellow. We see that the region red and yellow divides what are F G. A and B. Are. So the first region given here in orange or in red has A to B 0 to 1 where F of X. The higher function is X G of X is negative. X. We see the the area in yellow has G fx now swapped to the line x minus two. Thus the RV is two pi integral. 01, X x x x plus integral. Wonderful. Extrude X minus x minus two Dx. Are integral. Simplifies on the middle line which has anti derivative given here value that are anti derivative or rather anti derivatives for each of the two and two rules that are given bounds give 72 pi over five as our solution

Were given a set of curves in the line and were asked to find the exact volume of the solid obtained by rotating the region. Bounded by these curves about this line using a computer algebra system, the curves were given R y equals X. Why equals X times either the one minus sex over to and the line is, Why equals three. First, let's just sketch graph of the region. So this is going to be in the primarily 1st and 4th quadrants. I'm sorry, just really in the first quadrant. So we have the curve y equals X, which is simply a straight line through the origin in the 0.11 We also have the curve X times E to one minus X over to this has a zero at X equals zero, and to find where these two curves intersect, you want to set X equal to X E to the one minus X over to, in other words, either one minus x over two SD equal one, which implies that one minus X two equals zero. So the X is equal to two, so they intersect somewhere at the 0.22 and otherwise this function is going to look something like this. So the area that we're interested in is this region that I've drawn in red here, and we're going to rotate it about the line Y equals three, which is up here somewhere. Now we have that the volume of the solid formed a rotating about this line. Well, if we rotate about this line of solid we get has cross sections parallel to the Y axis, which are washers. The volume is going to be integral from X equals zero to x equals two of the area of these washers, which is pi times The large radius. This is our top function. Y equals three minus. The bottom function y equals X squared minus smaller radius, which is our top function y equals three minus the bottom function, which is X times E to the one minus X over to square the radius T X and using a computer algebra system like Wolfram Alfa, this is exactly equal to high times negative two e squared plus 24 e minus 142 3rd

All right in this problem we will discuss the region are bounded by the given courage And to utilize the shell method to find the volume of the solid generated by involving our about the Y axis. The curves bounding our region are Y equals X. Y equals two X and Y equals four. This question is channel your understanding of applications of integration namely the volume of a solid revolution shall not about the Y axis gives us the volume will be two pi integral. It'd be xfx managed dx dx where F X is greater than G X on a. To be so determine our A B F N G. We graph are so the region R is given here. We see that separated by two areas in yellow and orange. The area and yellow has F X equals two X and G of X equals X. The area in orange however has f X equals the line, Y equals four G of x equals X. Thus we integrate our region from 0 to 2 and 2 to 4 separately for this volume. This gives B equals two pi integral. 0 to 2 X two x minus x D x plus two pi integral to +24 X four minutes X dx simplifying these integral. And taking the integrative gives two pi x cubed over +3022 plus two, X minus x cubed over three from 2 to 4 were simplifies the volume 16 pie


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