5

1) Consider the region bounded by f(r) = Vinkz), z = 0 and y = 1 depicted below_FIGURE 1_Sketch and compute the volume of the solid that is generated when one revol...

Question

1) Consider the region bounded by f(r) = Vinkz), z = 0 and y = 1 depicted below_FIGURE 1_Sketch and compute the volume of the solid that is generated when one revolves the above region about the line y = 0.

1) Consider the region bounded by f(r) = Vinkz), z = 0 and y = 1 depicted below_ FIGURE 1_ Sketch and compute the volume of the solid that is generated when one revolves the above region about the line y = 0.



Answers

Consider the region bounded by $y=e^{x}$ the $x$ -axis, and the lines $x=0$ and $x=1 .$ Find the volume of the solid. The solid obtained by rotating the region about the horizontal line $y=-3$

Okay, So for this problem, we have forgiven the equations. Um, X equals y squared, and we're giving the equation X equals why? And we're looking for whenever it's revolved around y equals negative one. So if we look at this, we want to figure out where these two points intersect and we know that they're going. So when y squared and why equal each other, that is, whenever we have a zero in one. Okay, Anything else is not gonna be equivalent to its own square. And then, um, what we want to do is we wanna look at the equations. So because we have this why minus one This is now going to become, um when you first off change it toe white equal. So it's gonna be white, equals the square root effects. And because of the fact we have this negative one, it is now going to become one plus the square root of X, and this one is going to become one plus X. And so obviously we're going Thio handle this using the washer method, so it's gonna be pi with the integral from zero toe one, and then we're going to use the one plus square root of X square it as our outer radius and one plus x squared as my inner radius. Yeah, because the out the export one is is above the one plus X. So now I want to simplify these out. So it's going to be one plus two square root of X plus X minus one plus two X plus X squared. So now what I want to do is I want to change the side. So I'm gonna use a green to go through here so it's gonna be pi from 1 to 0. So I'm gonna have a one minus the ones that those canceled. I'm gonna have to square root of X minus, X minus X squared. And now I'm ready to integrate. So I have This is technically X to the one half power. So now I'm gonna have an X to the three halfs power, and I'm gonna have four thirds in front of it because I have to do two thirds times too minus one half X squared minus one third x to the third from 0 to 1. So this is going to be four thirds minus one half minus one third and then, of course, minus zeros. So this is going to be pie over to

Were given a set of curves and the line and rest to find the volume of the solid obtained by rotating the region. Bounded by these curves about this line curves your wife was X squared. X equals y squared. We're rotating about the line y equals one first. Let's just graph this region in the X y plane. So this is the region's bounded by two parabolas. So we have white was X squared was an upward facing parabola like this. We also have X equals y squared, which is a probably opening to the right this and so the region bounded by These is this slavery here. This is what we're going to rotate. And the line that were rotating about is the line y equals one above the region. So the graph the solid I'm really only going to be the first quadrant. You're amusing different scales on the x and Y axes. So this is what he solid looks like. If I draw Mm hmm Cross section. It looks like this so that it's clearly cross section is a washer or analysts with an inner radius of one minus. And then the top function is thes square root of X and an outer radius, which is one minus. And then the bottom function is X squared. So the area of this washer is pi times one minus X squared, squared minus pi times one minus square Root of X squared, which simplifies to high times one minus two X squared plus sex to the fourth minus one minus two square root of X plus x. We're simplifies even further. Two pi times x to the fourth minus two X squared minus X plus to root X, and therefore the volume of this solid is the integral from X equals 0 to 1 of area, which is three integral from 01 of pi times x to the fourth minus two X squared, minus X plus to root X e x and then taking it a derivatives. This is pi times 1/5 x to the fifth minus two thirds Execute minus one half X squared plus two times two thirds or four thirds x to the three halves from 0 to 1 and substituting this is pi Times 1/5 minus two thirds minus one half, plus four thirds. This simplifies to 11 30th times pi

Taking questions from the topic. Electronic contribution and the project. This question is asked that how many substances present, how many services are present In N. 2. 5. Shit Basically. For any quest to five shells. We know the common law finding number of assumptions. The number of sections can be determined by but the formula you know to and minus one. So If you try to point number of subsection, so if you'd like to find a good B0 to 5 -1 which is 0- four. So between 0-4 we have order numbers zero one, no and three. So basically how many options are there psychogenic formula? You can understand? The number of sales will be four which is so in And it was 2 5 shell Or in the 5th shall we have we have four substance present. Poor substance doesn't Prison. 1st 1 is exception because L. Is zero and the first one would be yes option. 2nd 1 will be This option because it is one that would be B section. 3rd 1 will be deception, reception and Portman will be exception. So in N equals 25 The number of substance are for you. Therefore, we can say that mm in fifth shell Or any kinds of five we have. We have four options the job? S being mhm I look Also unless forest 40 40 What? So these are the answer the question two you can see In the 5th channel we have for exceptions. four substance for us. 14, 14 what? And the formula that refuses all this question is Number of sources can be obtained by the value by the Cannula 02 in -1 zero 25. Management for 0 1, 2, B and B. Substance that we can have this particular.

All right, let's give us a picture of this. Yeah. Okay. Okay. There's the line. The X axis went X zero. Y is zero When X is one Why is 1 4th and when X is minus one? Wise 1/4 when X is too why is one? All right. Okay, X equals two. That's this line and Y equals zero. That's this line. All right. So here's the area or the region and we're gonna send it around the Y axis. So we have to cut it this way. It's always cut perpendicular to the axis of rotation. Okay, Since the region does not touch the axis of rotation, we're going to have a whole, okay, so There's the outside and then here's the inside of that one. All right. So the volume is the volume of the outside ring pie Big R squared H. When is the volume of the whole pie? Little r squared H. Or pie? Big R squared minus little R squared H. Okay. Okay. H is the height of the little cylinder? See me coloring it in which is the same as the thickness of the strip I made which is dy r is the distance from the axis of rotation out to the edge. So from here to there, so that is X on the right and X on the right is X equals two. And then little R. Is the distance from the axis of rotation to the whole so from here to here and that's X on the left and X on the left was oh we got y equals 1/4 X squared. So we've got to solve that for X. So for why is X squared? So X. Is plus her minus the square root of. For why? Okay so which one do we want? Well we want the one over on the right which is the positive square root? So the positive square root of four y. Or two times the square root of while. I'm just gonna leave it as four squared. So why? Because you want to square it anyway? Okay then we're gonna pile them up from uh sorry From y equals zero to this wide value here. So remember that were when X. Is too why is 1/4 2 squared? Which is one? So this point was to one So from 01 So the volume is Pi 01 two squared minus squared of four Y squared H. So Pi 014 -4. Y. Do you why pie for y minus four Y squared over two. So two y squared from 0 to 1 4 -2. So two pi


Similar Solved Questions

5 answers
Q} (10 marke] Sole the follouing diffctential cquation using Powtt scficI apptoach; uic nonzcro +D" (10 matks} Ut the ratio tegt I0 find tc open Iteral of conter ence and the rudiu: of convergence I0 te seres 4+, (*-6)" (10 marks)
Q} (10 marke] Sole the follouing diffctential cquation using Powtt scficI apptoach; uic nonzcro +D" (10 matks} Ut the ratio tegt I0 find tc open Iteral of conter ence and the rudiu: of convergence I0 te seres 4+, (*-6)" (10 marks)...
5 answers
Use IVT to show that the curve y 22 and y = cOS € intersect on the interval [0 , w]:Use bisection method twice to narrow the interval to length w/4.
Use IVT to show that the curve y 22 and y = cOS € intersect on the interval [0 , w]: Use bisection method twice to narrow the interval to length w/4....
1 answers
Specify which of these salts will undergo hydrolysis: $\mathrm{KF}, \mathrm{NaNO}_{3}, \mathrm{NH}_{4} \mathrm{NO}_{2}, \mathrm{MgSO}_{4}, \mathrm{KCN}, \mathrm{C}_{6} \mathrm{H}_{5} \mathrm{COONa}$ $\mathrm{RbI}, \mathrm{Na}_{2} \mathrm{CO}_{3}, \mathrm{CaCl}_{2}, \mathrm{HCOOK}$.
Specify which of these salts will undergo hydrolysis: $\mathrm{KF}, \mathrm{NaNO}_{3}, \mathrm{NH}_{4} \mathrm{NO}_{2}, \mathrm{MgSO}_{4}, \mathrm{KCN}, \mathrm{C}_{6} \mathrm{H}_{5} \mathrm{COONa}$ $\mathrm{RbI}, \mathrm{Na}_{2} \mathrm{CO}_{3}, \mathrm{CaCl}_{2}, \mathrm{HCOOK}$....
5 answers
Adjustment of the lens to focus on objects close to the viewer is calleda. convergence.b. visual accommodation.c. focusing.d. constriction.
Adjustment of the lens to focus on objects close to the viewer is called a. convergence. b. visual accommodation. c. focusing. d. constriction....
5 answers
Prepare your data sheet (spreadsheet) by following steps 5 and6 below. Then come back to stepWatch the following video about the Experiment Overview (pressctrl + click on link).Press ctrl + click on any of the following links and follow theinstructions stated on step 4.Run#1 Run#2 Run#3 Run#4 Run #5Watch and gather data from the chosen data run above, makingsure you:Take note of the background counts.Only record the data from the PMT de
Prepare your data sheet (spreadsheet) by following steps 5 and 6 below. Then come back to step Watch the following video about the Experiment Overview (press ctrl + click on link). Press ctrl + click on any of the following links and follow the instructions stated on step 4. Run #1 R...
5 answers
Find the complementary solution Xc for the ODE in (a):[4 marks]Find the particular solution Xp for the ODE in (a).[6 marks]Derive the ODE that describes the time evolution of temperature T inside the tank: [6 marks]
Find the complementary solution Xc for the ODE in (a): [4 marks] Find the particular solution Xp for the ODE in (a). [6 marks] Derive the ODE that describes the time evolution of temperature T inside the tank: [6 marks]...
5 answers
Consider the vibration of a gaseous iodine molecule,I2, with force constant k = 172 N/m.(a) Calculate the ground state energy (ineV) of this molecule.(b) Is the molecule infrared active?Explain. Is the molecule rotationally active? Explain.
Consider the vibration of a gaseous iodine molecule, I2, with force constant k = 172 N/m. (a) Calculate the ground state energy (in eV) of this molecule. (b) Is the molecule infrared active? Explain. Is the molecule rotationally active? Explain....
5 answers
S%f uphill grade. This ineans that the slope of the road You ae driving on Toad that has Approximate the amount of vertical change in your position if you drive 400 feet:100
s%f uphill grade. This ineans that the slope of the road You ae driving on Toad that has Approximate the amount of vertical change in your position if you drive 400 feet: 100...
5 answers
Use the graph ofy=f (x) to the nght t0 discuss the graph = of y ((x) Organize your conclusions in a tableDetermine how the properties on different intervals of {' (x) affect f(x) Complete the table below:O<x<andand32-474<1<0andand0<x<4andandX=44<x<oandand20
Use the graph ofy=f (x) to the nght t0 discuss the graph = of y ((x) Organize your conclusions in a table Determine how the properties on different intervals of {' (x) affect f(x) Complete the table below: O<x< and and 32-4 74<1<0 and and 0<x<4 and and X=4 4<x<o and and...
5 answers
1. (18 points) Determine if the following limits exist: In each case prove and explain your argument.(a)lim x3(15*+y4ry2 lim 1'(0,0) x2+y2lim 17(0,0) 1+310
1. (18 points) Determine if the following limits exist: In each case prove and explain your argument. (a) lim x3(15*+y 4ry2 lim 1'(0,0) x2+y2 lim 17(0,0) 1+310...
4 answers
Areoumaey hon Gh Fenaon inkJ # (equbed lo plale = Cu3 5ge/As)[-27 | ey elctrolysis hom an ANO3lglaq) solution? Assume thal the vollage is 10. Volts 3754415,
Areoumaey hon Gh Fenaon inkJ # (equbed lo plale = Cu3 5ge/As)[-27 | ey elctrolysis hom an ANO3lglaq) solution? Assume thal the vollage is 10. Volts 37544 15,...
5 answers
For" any set 4; 4 U @U.TrucFalsc
For" any set 4; 4 U @ U. Truc Falsc...
5 answers
Item3 0 VP Oee wilh u = 2 thal nas 5 comploting tis problem I 1 1 3 (598 V 1L Part B Submlt AG Exprosryouranswer Ujino answtr uainontuo 1 undo 3 1 urido rosa/ 3 significant fiqures. re5 1 1 AuybQuutotu 3 38C
Item 3 0 VP Oee wilh u = 2 thal nas 5 comploting tis problem I 1 1 3 (598 V 1 L Part B Submlt AG Exprosryouranswer Ujino answtr uainontuo 1 undo 3 1 urido rosa/ 3 significant fiqures. re5 1 1 AuybQuutotu 3 38C...
4 answers
Round {0 the nearest cent V interest ~did Speedy rate was 5.50% compounded Movers borrow for a debt that accumulated to Questlon 6 of 12 semi-annually 554,317.74 in five#[email protected] @ 0 1
Round {0 the nearest cent V interest ~did Speedy rate was 5.50% compounded Movers borrow for a debt that accumulated to Questlon 6 of 12 semi-annually 554,317.74 in five # 0 0 @ @ 0 1...

-- 0.021124--