5

Find the volume tne solid genera udrevotving the shaded region about Y-arisTha voluma 54* - cubic rt; (Typa en exact answ0r, using *a8 noodod )...

Question

Find the volume tne solid genera udrevotving the shaded region about Y-arisTha voluma 54* - cubic rt; (Typa en exact answ0r, using *a8 noodod )

Find the volume tne solid genera ud revotving the shaded region about Y-aris Tha voluma 54* - cubic rt; (Typa en exact answ0r, using *a8 noodod )



Answers

Find the volume of the solid that results when the shaded region is revolved about the indicated axis. (Graph cannot copy)

Okay, so for this one, we're going to have a graph. I'm just going to kind of quickly sketch it. So we're gonna have where X is. Ah, boundary of one. And then, of course, along the axis at X equals zero. And we're gonna have to minus, um X squared. And we're also gonna have y equals X. So I know that why equals X is going toe Look, something kind of like this. And then, of course, why equals two minus X squared is going toe Look, something kind of like this eso we're looking at the area that is going to be here in this little section. So what I want to do is I want to use theme washer method and just to quickly review with the washer method formula. Looks like it's the volume equals pi from a to B of f of X squared minus G of X squared and then, of course, all in terms of DX. So what I want to dio is I'm going to set up my two minus x square to be my my top function. And then my y equals X to be my GF X function So I'm gonna have equals pi from 1 to 0 of two minus X squared, squared minus X squared to be squared d x I'm sorry, X squared. So then I'm gonna have pie and I want to simplify this down, so I'm gonna have four minus four x squared plus X to the fourth minus x squared. So this is going to be four minus five X squared plus X to the fourth, all in terms of DX. So now I can go ahead and integrate, so I'm gonna have four X minus five thirds X to the third plus 1/5 X to the fifth from 1 to 0. And so this is going to be pie. And then, of course, this is gonna be four minus five thirds plus 1/5. And then, of course, of zeros with all of these in terms of excess, all of those air going to be zero. So if I were to combine all these and simplify, I'm gonna get 38 pi over 15

Okay, So for this problem, we're gonna have our crafts and we're gonna have y equals two. And then, of course, the axes at Y equals zero. And then I'm also going toe have the axis here, which is X equals zero. And then I'm going toe also have the axis. It's gonna look something kind of like this. It is just a approximation sketch is gonna be y equals, um and it was three minus two x. Okay, so we're essentially finding the area of the shaded region in here. And so for this one, I'm going to use the disk method, and that formula is volume equals pi times the integral from A to B of f x squared DX. So what I want to do is I'm gonna set this up. Is being pie from the integral of 0 to 2 and this is going to be so What I want to do is, since I'm doing this in terms of the Y Axis is, I do want to take this Why equals and do this in terms of X, so X would equal three minus y over to. So now what I want to do is I want to, um, put this into the three minus y over to squared d y. So if I were to simplify this, this is going to be nine minus six y plus y squared all over four. Now what I want to do just to make this easier, I'm gonna move this out toe thio with the pie, and this is going to be integrated nine y minus three y squared, plus one third whites of the third, all in terms of 0 to 2. So if I were to go through, I've got pi over four, and then I'm gonna have 18 minus 12 plus eight thirds minus zero. And then, of course, this is all going to be 26 3rd. I can simplify this down to being 13 pi over six.

Okay, So for this particular problem, we're bounds by, um, where we have X equals two and we have why equals two and then we're actually bound by a graph that kind of comes in like this, which, of course, is why equals one over X. So what we want to do is we wanna look at the area kind of in this little section, and it is, of course, going across the y axis. So what I want to do is I want to set up the we're going. I'm going to use the washer method because it's in between two different functions. So it's gonna be volume equals pi times the integral from A to B of f of X squared minus G of X squared D X. And so I'm gonna have pie, and then I'm going to use. Obviously, we have the, um, the two being one of them and then this bought here is going to be at one half, because excess too. So it's gonna be why is one half and I'm going to use The first function is being Why equals two. I'm sorry, X equals two, and then I'm going to do this in terms of X equals. So I can change this toe X equals one over why? Which is just the inverse of it. And this will be in terms of d y since we're doing across the y axis. So I'm gonna go ahead and simplify this out as much as I can. So it's going to be four minus one over y squared and all of this in terms of d y. So then it's gonna be pi times for why and then negative one over y squared will become positive one over white in terms of two and one half. So this is going to be pie and then eight plus one half minus two plus two, which is going to be nine pie over to.

Okay, So for this particular graph we're looking at, um, the interval between pi over or excess pi over four and pi over two. And we're looking at the co sign. I'm sorry. Square root of CO sin of X and the axis. So we're looking at this interval here, So I want to use the disk method, which, just as a reminder is theme is pi times the integral of a to be and then times f of x squared d x, This is in terms of the X axis, so we're gonna leave it in terms of X. So it's gonna be pi from at times integral from pi over 42 pi over two of the square root of co sign of X squared DX. So obviously, because we're squaring a square root, they disappear, they cancel each other out. So I'm looking at the integral of co sin of X dx, which is going to be sign of X from pi over 42 pi over two. So this is going to be pi times one minus the square root of 2/2. And so I can simplify this down to being to minus the square root of two all over two pi


Similar Solved Questions

5 answers
A school principal plans to form teams from 252 third-graders, 420 fourth-graders, and 378 fifth-graders so that there is the same number of students from each grade level on each team. Ifall students participate, what is the largest possible number of teams and how many students will there be on each team? (10 points)
A school principal plans to form teams from 252 third-graders, 420 fourth-graders, and 378 fifth-graders so that there is the same number of students from each grade level on each team. Ifall students participate, what is the largest possible number of teams and how many students will there be on ea...
5 answers
Use the Root Test to determine whether the series converges or diverges: (k)
Use the Root Test to determine whether the series converges or diverges: (k)...
5 answers
7. Evaluate (5 Points) cos( Jr + a)ev dx(sin(a + Jr) cos(a + W)) ex + €(sin(a + JI) + cos(a + JT)) + €(tan(a + JF) + cos(a + m)) e + C(sin(a + VF) + tan(a + W) ev + €(sin(a + VT) + cos(a + ,T))e + €Submit
7. Evaluate (5 Points) cos( Jr + a)ev dx (sin(a + Jr) cos(a + W)) ex + € (sin(a + JI) + cos(a + JT)) + € (tan(a + JF) + cos(a + m)) e + C (sin(a + VF) + tan(a + W) ev + € (sin(a + VT) + cos(a + ,T))e + € Submit...
5 answers
Poinl) Find equalians ot Iht Langjent plane and normal Ilne to the sutface Tangeni Pidng: (Make (hu cooflcieni equallora" cut = 0t tho polnt (19 ,Nolmal Iina (I0
poinl) Find equalians ot Iht Langjent plane and normal Ilne to the sutface Tangeni Pidng: (Make (hu cooflcieni equallo ra" cut = 0t tho polnt (19 , Nolmal Iina (I0...
5 answers
~6(1 point) Let f(a) 26 _ 7 8 f' (x). f' (x) =Find
~6 (1 point) Let f(a) 26 _ 7 8 f' (x). f' (x) = Find...
5 answers
And columns using 400 feet = fencing: What dimensions will maximize the totalpoint) armo builds rectangular grid of pens with aroa pon? The total width each rOw of the pens should b0 100 tho total holght ot each column pons should b0 which glves the maximum area of 5000fcet ,Icot,square (eet:
and columns using 400 feet = fencing: What dimensions will maximize the total point) armo builds rectangular grid of pens with aroa pon? The total width each rOw of the pens should b0 100 tho total holght ot each column pons should b0 which glves the maximum area of 5000 fcet , Icot, square (eet:...
5 answers
Using enthalpies of formation (Appendix C), calculate $Delta H^{circ}$ for the following reaction at $25^{circ} mathrm{C}$. Also calculate $Delta S^{circ}$ for this reaction from standard entropies at $25^{circ} mathrm{C}$. Use these values to calculate $Delta G^{circ}$ for the reaction at this temperature.$$mathrm{C}_{3} mathrm{H}_{8}(g)+5 mathrm{O}_{2}(g) longrightarrow 3 mathrm{CO}_{2}(g)+4 mathrm{H}_{2} mathrm{O}(g)$$
Using enthalpies of formation (Appendix C), calculate $Delta H^{circ}$ for the following reaction at $25^{circ} mathrm{C}$. Also calculate $Delta S^{circ}$ for this reaction from standard entropies at $25^{circ} mathrm{C}$. Use these values to calculate $Delta G^{circ}$ for the reaction at this temp...
5 answers
Find all the currents and voltages across each resistor and cell in the following circuits;18 V12 V6 V
Find all the currents and voltages across each resistor and cell in the following circuits; 18 V 12 V 6 V...
5 answers
What would be the theoretical freezing point of = water? The freezing point constant of water solution made by dissolving 14.09 g ol MgSo 4 in 160.0 9ot Clmolal. 0.3999 "C 22.501 2.501 'C ~0.3999 -1.450 'C4Rblal
What would be the theoretical freezing point of = water? The freezing point constant of water solution made by dissolving 14.09 g ol MgSo 4 in 160.0 9ot Clmolal. 0.3999 "C 22.501 2.501 'C ~0.3999 -1.450 'C 4Rblal...
5 answers
Oketching Quadratic Surfaces: Sketch the following quadric surfaces. Put the equation for the surface in stan- dard form and identify the quadric surface.28. 16y2 422 + 1622 = 0 30. 4 8y? 822 = 0 32 _ 4r2 + 16y2 + 6422 =64 34. 4y2 1622 _ 1622 = 64
oketching Quadratic Surfaces: Sketch the following quadric surfaces. Put the equation for the surface in stan- dard form and identify the quadric surface. 28. 16y2 422 + 1622 = 0 30. 4 8y? 822 = 0 32 _ 4r2 + 16y2 + 6422 =64 34. 4y2 1622 _ 1622 = 64...
5 answers
Ealance the following redox reaction in an acidic solution: How many Warer molecules are in the balark2d reae3on? NO; 11l NQSelect the correrr answer below:
Ealance the following redox reaction in an acidic solution: How many Warer molecules are in the balark2d reae3on? NO; 11l NQ Select the correrr answer below:...
5 answers
Circle orums Submit 1 H 1 whose Find Ii question Dasibtempts. y-coordinate is H I7 forum on the unit 7 (Enter Preriew: (Enter cucle multiple multiple answers answers separated by separated by commas commas
Circle orums Submit 1 H 1 whose Find Ii question Dasibtempts. y-coordinate is H I7 forum on the unit 7 (Enter Preriew: (Enter cucle multiple multiple answers answers separated by separated by commas commas...
5 answers
In a transmembrane protein like the one shown, where would youexpect to find a hydrophobic amino acid like valine? Select one or more:a. none of the aboveb. anywhere in the protein, with equal probabilityc. on the exterior surface of the protein, interacting withwaterd. in the transmembrane portion interacting with lipidfatty acid chainse. in the interior of the folded protein, away from themembrane facing other amino acids.
In a transmembrane protein like the one shown, where would you expect to find a hydrophobic amino acid like valine? Select one or more: a. none of the above b. anywhere in the protein, with equal probability c. on the exterior surface of the protein, interacting with water d. in the transmembrane ...
5 answers
If 8.02 mol of NOz occupy 18.7 L how many liters will 4.42 mol of NOz occupy at the same tempcrature and pressure?
If 8.02 mol of NOz occupy 18.7 L how many liters will 4.42 mol of NOz occupy at the same tempcrature and pressure?...
5 answers
LCl V = Tu Find the angle (measured in radians) between u = [ -] 2 5 ] and v = [ 2 -3 1.6]; Find the magnitude of < -1,2,87, <4.0.5,3 > and 1.2i - 3j +L.lk; Find the cross product < 0.5,2,3 > x <3,6,9>: Show that the vector produced by multiplying v by the reciprocal Let v be a vector in Rn. Vertat (i e has magnitude
LCl V = Tu Find the angle (measured in radians) between u = [ -] 2 5 ] and v = [ 2 -3 1.6]; Find the magnitude of < -1,2,87, <4.0.5,3 > and 1.2i - 3j +L.lk; Find the cross product < 0.5,2,3 > x <3,6,9>: Show that the vector produced by multiplying v by the reciprocal Let v be a ...

-- 0.020786--