solve-the-given-problems-by-integration-find-the-volume-generated-if-the-region-of-exercise-30-is-re

Question

Solve the given problems by integration.
Find the volume generated if the region of Exercise 30 is revolved about the $y$ -axis.

Foster Wisusik · Accepted Answer

All right, This question wants us to revolve this region around the y axis. So first, let&#x27;s look at the slices we&#x27;re gonna have to make to complete this. So looks like we have two different behaviors going on throughout this region. So between zero and two, as you see, all slices have the same with the distance from the from the axis of revolution remains constant. So where? Why in the range zero and two, the radius just equals two. And you can see that we&#x27;re only dealing with one radius here because this is touching the access of revolution. Then after you get past two, you see that our slice depends on the blue curve also. So, using our rule of top minus bottom done from 2 to 6, the radius is two minus our function of why. And in this case, since why equals X squared plus one, we can solve for insult for acts, which means that X equals square root. The positive one of why minus two, this should be a two. All right, right. And that&#x27;s our f of y. So now let&#x27;s find the volume. Since we have two different behaviors will have two different girls, so v equals pi times the integral from 0 to 2 of the radius squared, which is to plus high times theater girl from 2 to 6 of our second radius to minus square root of why minus two quantity squared. Do you? Why and plugging this into a calculator Using the F in integrate function, we find that this turns out to be 32 over three times pi.

Adnan Gill · Answer

that&#x27;s not solving this question by visualizing the triangle. So the triangle has a point. B zero. It has a 00.0 H, and it has the 0.0 as it&#x27;s your disease. Let&#x27;s Prodi lined the partners here. The equation of this line is why is equal to negative h o R B x. I want to rotate this volume around the X axis, so I will take a vertical elemental piece off thickness D X and the radius off That would be equal to why so I can save D V equals pi r squared DX or pi times negative h over b X squared DX, which equals pi times each square over B squared X quick. And in order to find the volume, I just need to take the integral off X squared from zero to be and times that with pie, it&#x27;s great be squid. So the integral off eight square is H. Q. Ball off the middle of X squared is excused Dorothy, so the value will be equal to pi. Times eight squared over B squared and then Times X cubed over three, and the liberties are from zero to be well, that would be equal to pi. Times eight squared over B square and then times be cubed over three minus you, which simplifies to pi over three h squared. Be so the volume of solid generated when this area is rotated about the X axis would be pie or three times eight squared over B. No. The next part wants me to take the same area and rotated about the y axis so I will still have the same line. Why is equal to negative H over B X? But now I want to write it as X is equal to negative. Be over h Why? And the reason is because now I will take my elemental piece to be horizontal on the thickness off would be D y. So I would say TV is equal to pi r squared d y and why would actually be equal to X. So this would be pie times negative be over h why squared d Y or attack would be equal to pi. Times B squared over each squared y squared d y. And let&#x27;s find the volume. The value would be pie. Be square over h squared in the grill off. Why square G y from zero to H and that while you were people to pi times p squared over h squared and the integral off y squared is why cubed or three? The limits are from zero to h. So that would be pi times p squared over h squared times each cube over free minus Europe which simplifies to pi times p squared each over threes of the one What? This area is rotated a party. Why access? Ask you in, but I or three b squared h

Amy Jiang · Answer

Okay, so we have wise to 3/4. Why? He could know Exiting good to go and exited. Good too. This is rotated around the Y axis. So why so physical? Three. Why add to its one at exit vehicle? To a bit of why minus three. We&#x27;re reading this secret. You in here? So we get the integral from 1 to 3 of pi 31 His wife squared the wife plus the integral from Joe to one of high two squared D white. Okay, so we get negative pie time stream when it&#x27;s why Huge over three. Evaluated at three and one for pie. Evaluated at one in jail. That&#x27;s before pipe. So yet H over three five plus four. Pi is equal to 20 play over three.

Adrian Fan · Answer

for this problem or asked to calculate the volume of Ah, solid of revolution and the solid is described as follows. Go to take the graph of the line. Why equals three minus X? Take the region bounded by that line as well as the lines. Why equals zero? It&#x27;s going to be the X axis and the lines X equals zero to the y axis and the line X equals True. So it was going to be This trap is oil region and then we&#x27;re going to rotate around the y axis and we want to find the volume of the results you saw. The revolution should be revolved like this. For this problem, I&#x27;m going to use a method of cylindrical shells. Briefly summarize that method. Want to take a look at very small rectangle? Look here, arbitrary ex co ordinate and then consider the cylindrical shell form. When we rotate that rectangle around the y axis, the result is going to be a cynical shell whose area is too high. X times three minus X because the length of this segment is three minus x, so that is one cross sectional area. One cylindrical show. Now, if we take the integral off this cross sectional area function over the Range X equals 0 to 2. We will obtain the total volume of the salt of revolution. And I&#x27;m going to show you how this integration is done. The volume of the salt of revolution is going to be integral. That cross sectional area function. That&#x27;s true. Hi X Times quantity three minus x The DX. Now I want to evaluate this integral first factor at the troop hi. And then we&#x27;ll expand this part with the distributive property to get three x minus X squared DX and then take the anti derivative of that as the application of the power rule. It&#x27;s going to give us three x squared over true minus X, cubed over three. Then after it, take the next train of that between zero and truth. Continue my calculations there. The rest is just arithmetic. It&#x27;s true. Pi times Well, three times two squared over True. That&#x27;s three times too. Or just six quietness Too cute over three. That&#x27;s eight over three on Dhe. If you do the arithmetic here were true. Pi times 10 over three, which is 20 pile birth three and that is our final answer

Caelan Thomas · Answer

First, we&#x27;re going to want to find our bounds of integration by taking this equation and substituting in X equals zero. And so we&#x27;ll get zero plus Why minus one squared equals one, which is going to give us y minus one equals positive. One or Y minus one equals negative one. And so we get Why equals two or why equals zero. So we&#x27;re integrating from 0 to 2 and we&#x27;ll take slicing to get the area. Pi r squared. Where are it&#x27;s going to be equal to the square root of one minus y minus one squared And if we square that we lose the square root So we have the integral of one minus Well, I minus one squared de y, and that&#x27;s going to give us the integral of why minus or one minus. Why squared? Minus two y plus when which will simplify, too. The integral of negative y squared plus two way, and now we&#x27;re ready to integrate. So when we integrate, we get pi times native 1/3. Why cubed plus y squared evaluated from 0 to 2, and if we plug in two and zero, we get Pyatt times negative 1/3 times two cubed plus two squared minus zero Which gives us pi times negative. 8/3 plus four which is 12/3. And so we get high times 4/3 where, if you like for pie over three for the volume.

Problem

Solve the given problems by integration. Find the…

Problem 31 Easy Difficulty

Solve the given problems by integration.
Find the volume generated if the region of Exercise 30 is revolved about the $y$ -axis.

Answer

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Basic Technical Mathematics with Calculus

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Allyn J. Washington, Richard S. EvansBasic Technical Mathematics with Calculus

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Basic Technical Mathematics with Calculus