5

6. Let B be the solid in the first octant hounded by 2 = 4- 22 v ,y = I 2 = 0, and y = 0. Set up an integral in cylindrical coordinates that calculates the voluue o...

Question

6. Let B be the solid in the first octant hounded by 2 = 4- 22 v ,y = I 2 = 0, and y = 0. Set up an integral in cylindrical coordinates that calculates the voluue of &.

6. Let B be the solid in the first octant hounded by 2 = 4- 22 v ,y = I 2 = 0, and y = 0. Set up an integral in cylindrical coordinates that calculates the voluue of &.



Answers

Let $G$ be the solid in the first octant bounded by the sphere $x^{2}+y^{2}+z^{2}=4$ and the coordinate planes. Evaluate $$ \iint_{G} \int x y z d V $$ (a) using rectangular coordinates (b) using cylindrical coordinates (c) using spherical coordinates.

Okay, So this problem wants you to evaluate the triple integral of exposed wide Losey over the region off E. Where is is a solid that is, in the first Occident and is below the pair up. And it is below the probable oId Z equals four minus x squared, plus y the evil four minus x four miles y squared. All right, so the first thing we do is to convert everything into cylindrical coordinates. So if we were to convert our into grand into political coordinates, we would have extra swipe a Z where X is our co sign down our coastline. Dana, we have Why is our side data our scientist? No. And we have the just the and D V in cylindrical cornices are times are times DZ, pr di fada. All right, so now let's look at the boundaries. Yeah, in the first. Often that means our X y and Z components are all positive. Traveling really mattering cornets and our problem oId z before my eyes expire, Biswas, where will that can be rewritten as e equals two or minus r squared? And this is a downward facing problem. So looking at our boundaries versus DZ. So the bottom of our Z is the X Y is the X Y plane because I remember one of our arm. It is in the first octave. So x Y plane is The downward plane is a plane of Z equals zero Essentially, that restricts the region. Then we have the downward facing parable oId Ziegel. Four mice are squared, which is the upward restriction. So we could just let in before my eyes are square. Yes, we have the are So er and this obviously this largest one c equals zero because is done, we're basing problem. So we just said 0 to 4 minus r squared we can get our off to So our radius goes from 0 to 2 and finally our angle theta d'hara. It goes from zero two pi house. Now why is it pie house? Well, it's pie has because is in the first oxygen. So when the X y plane, the X and Y components have to be positive. All right, so now that we've got everything is little coordinates and we've got our boundaries now it's time to calculate it. Okay, so let's first integrate this with respect to z So our boundaries or the R and D d and I say the same. So if we integrated with respect Dizzy Well, I knew it. First get R squared the equal sign data our squares equals and data plus our square and the science data whispers he signed via was 1/2 rz sward 1/2 our a z squared And the boundaries are, of course, from four minus R squared from 0 to 4 minus are square and the other two or d r d Fate up. Okay, Next if we plot If we were to plug in the points for Mayes R squared and zero into our into ground would get the double into grown off double integral of R squared times four minus R squared. Course I do. You know waas R squared. I was four minus r squared. Sorry, they are waas. Well, it's 1/2 are times four minus R squared squared DRD data. Okay, so now we have to integrate. Now we have to integrate this with respect to D I or d thing up. Well, looking at this equation, you can probably tell that this would be This will be a struggle to integrate with respect to our So how about we try integrating it with respect to defeat data first? So let's swap the boundaries to detail. They are because if we did, with respect to D data and, er we're bound to get since we have a co sign and a sign when we're dealing with a region boundary a pirate, too. We're about to get a zero somewhere, so we'll probably end up most nicer than if we just did this ugly Integral will just back to our So if we were to integrate this with respect to data would have an integral from 0 to 2 to the boundaries of our we would get into grown r squared. I was four miners r squared. I'm silent. Data minus R squared times four minus R squared coastline data plus 1/2 one house are times for monies. R squared squared times stayed up and this is from pi over two Priore from zero to Pi Overton Oh, respected data with respect to a d. R. So now if we were to plug in our boundaries, uh, from zero department tomb, we would end up with an integral in the winter grown off of our square Four minutes or square four miners are swear plus pi over four times are times four minus R squared, squared, plus are square All right, four minus R squared, squared because off the coastline so would be at this. Oh, yeah, and respect the d r When were you just being combined? These two terms. And so we were to combine those term we could get a simpler into grow from 2 to 00 to our square I was four minus are square plus pi Ford's Our times four minus R squared, squared Respect the d r. All right. And if we were to expand this out so we can further simple fire we get an integral you go from two from zero to off Oh, yeah and be distributed to r squared into parenthesis east into our before my eyes are square Viet eight R squared When it's to our to the fourth to our to fourth plus pi over four our times 16 are minus eight R Q plus our bit to the fifth power Do you are? And now that we have this, we can integrate with respect to our to get to get in. Over three are cute. Minus to are to the fifth over. Five plus hi over four times IMEs times eight 18 r Squared. When is to argue before? Plus our six over six from 2 to 0. Okay. Wait. I see my hair here. This are not supposed to be there. I already distributed the are. So if now if we're simply plug in the values from zero to, we get a final answer of woman 28 over 15 plus a pi over three. And this is your final answer.

All right. So you want to go ahead and solve the Portugal already of X plus Y plus E. T. V. Yeah but it is just uh the region in the first act in yeah, bounded by the probable. Oid Z is equal to four minus X squared minus Y squared. Mhm. Now we know that the first accident Gives us two bounds on our payroll. So we know that it means that Z. is greater than zero and that data must be between Pi over two and 0. And to find your balance and arm we can just said um are tabloid equation which in cylindrical coordinates is for minus R squared Equal to zero. Since we know that's are found. That's the x. Um the bottom surface that we're talking about. So this just gives us R squared equal four or articles to. So then our last um val dishes that ours between zero and two. So you can go ahead and convert this entire thing. The cylindrical coordinates knowing that devious shifts are easy t already data. So this just ends up giving us It's a girl from 0 to Pi I'm sorry not to brian pi over two. It's a girl from it's here to to to go from 0 to 4 minus R. Squared of our coastline data. Which is X. In cylindrical coordinates R. Plus are assigned data. Which is why. And let's see our dizzy. Do you already paid up? Yeah let's go ahead and drag this interview down here and we can go ahead and multiply it out with ST bounds giving us our square times co sign data. Scientific data plus a Harzi T Z. D. Already data. Right? So the first thing you can do is integrate with respect to Z. Okay. Mhm. Which should give us since this does not have a Z. And this does we can just go ahead and plug in our bounds immediately. So we know that this term will just live. Leave us with um Z in front and then this term we'll just leave us with as you squared over two. So if we go ahead and plug in our bounds we'll see that. Um This first term becomes R squared co signed data for science data. Mhm. Times four minus R squared and then zero gives us zero. So you have to worry about that. And then the second term gives us our times four minus R squared squared all over two. And then again 00 So we don't have to worry about that. Do you already data? Mhm. And then we can go ahead and simplify this further. We can go ahead and multiply both out. So leaving us with are you sorry for R squared minus hearts? The fourth could decide data. Side data. Okay. Plus and then we're gonna go ahead and square this in it. Let's apply in the yard at the same time. So we have All over two. So this gives us um 16. We'll go ahead and do the R. After. So we have 16 -8 R Squared Plus. Hard to the 4th. And then if you go ahead and put the the So the R. N. We have 16. Our last eight are acute. It's hard to the fifth. The arctic data. So now we can just integrate respect to our giving us the following. So we have 4/3 R cubed minus hearts to fit over five times. Co science data. Science data. Oh and then we can do this on our hands. So this this term becomes a it R. Which will give us for R squared. This term becomes a four hour cube which gives us minus are to the fourth and this term gives us Artists 6/10 or 12 from Jared to do data. Now we can go ahead and plug in our bounds. We'll see that zero gives us zero for all the ours. So we can go ahead and just see that this is four times R cubed. So 24 times 8/3 32/3 -32/5. Because I'm fada was signed data plus We have 16 -16 plus 64/12. You pay them This cancels 64/12. I just equal to 16 over um three. So then we have It's a girl from 0 to Pi retune 32/30 minus 32 or five and go ahead and do that on the side. So we have 32 times 1 3rd 25th Which is equal to 32 times a five -3/15 Or 64/15. So we have 64/15. Co science data science data plus 8/3 are sorry 16/3 you data. And then we can go ahead and integrate this. Finally giving us 64 15 scientific data minus coastline data Plus 16/3. They don't From 0 to Pi over two. Okay. And then we can go ahead and evaluate this 64 or 15 times will separate both. So this is the left parts. We have sine of pi over two which is one minus cosine of power to zero and then subtracting sign of zero which is zero. And then subtracting my minus cosign theta, tau and Kazan zeros once we have this and then plus 16/3 times prior to This is just one plus one or two. So we have 128 over 15 to us eight thirds five as our final answer

For giving an integral. We were asked to evaluate Simple and the role of X plus y plus C off the region K, where he is the solid in the first, often that lies under the tabloids. Ecause four minus X squared minus Weisberg While this parable Lloyd Z equals four minus X squared minus y squared, this intersect the X Y plane, which is the plane Z Po zero in a circle X squared plus y squared equals four or influential coordinates. This is R squared equals four, and then a Zara's positive. This implies articles, too, and so and so into a coordinates. Our region E If you set of triples our data Z, it's the fatal eyes. Well, because when the first often realize between zero empire but to are is going to lie between zero and two and Z will lie between zero The X Y plane and between the Paraiba Lloyd four minus X squared minus y squared. Reaching rectangular, rectangular or mixed. Switched to yeah, political coordinates is four minus R squared. And so the triple Integral um, experts y plus z over the region E. This is the iterated integral, which is integral from 80 to pi over two and drop Marco 02 in the Balkans equals zero for minus R squared of a function in terms of cylindrical coordinates. So this is our cosine theta plus our sign data plus Z 20 differential, Which for cylindrical coordinates, this is our times Easy BRD data within the anti derivative with respect to Z we get and they grow from zero to pi over to you go from 0 to 2 one back going out on our hold Arkan's ours are squared co sign data plus sign data eyes mm plus one half times are times Z squared from Z equals zero To see those four minus r squared DRD data with the anti derivative Sorry. Evaluating integral from zero to pi over two integral from zero to and this is or are squared minus art of the fourth Times Co sign the A plus sign data plus one half times are times or minus r squared, squared DRD theater and taking the anti derivative with respect to our this is integral from zero to pi Over two of here this is four thirds are cute minus 1/5 part of the fifth times Co. Sign data plus sign data and then this next term. Do the use institution in your head We get, see one half times negative one half times one third This is negative 1 12 times the inter function or minus r squared to the new power three from r equals 02 Data and evaluating you get integral from zero to pi over two and then plugging in. This is 64 15th times the cosine of data plus the sign of data and then plus and 12 times for cute, which is 16 thirds deep. Data taking anti derivative with respective data. This is 64 15th times, man man, today of your fear is going to be signed data minus cosign data. A 16 3rd data data equals zero to pi over two. Evaluating you get 64 over 15 times one minus zero plus 60 16 3rd times pi over two minus 64 15th times zero minus one minus 63rd time. Zero because zero and we simplifies to two times 64 15th 1 28 15th plus a thirds pi

In the question we have to use the devil. Integration could find the volume of this which is in the first off pain and is wanted by a parable. Life off Zen Nichols to excess with class by sweat. And this rain that exists where my square pulse Before now moving works This solution the general configuration is declared off Circle of Radius to using the shell matter integration from zero people by my lex and grew at six. Where the X it was too high. By who? By four upon by now using the double integral which is integration from zero report go integration from 04 minus x the square heads square last. Why Square day by the express dissolve in the inner integral. Really? That regulation from zero war on three and four minus X s wearing who? Excess Where? Less way less No. Yes, some sort of this We really get work on bringing one upon four x and two x two square first, two mental minus excess for less signing waas off and spy to them. It went from zero to solving this to buy, which will really find a lot. Thank you


Similar Solved Questions

5 answers
0 8 Homework: 1 SectionEsta 1111 1 1,czn1
0 8 Homework: 1 Section Esta 1 1 1 1 1 1,czn 1...
5 answers
41.66 poirts Suppose A = [az . ..a;] and B = [bt bi] are two 4 x 4 matrices so thatAB =12What is Abz That is. what is A times the second column of B? Not enough information to tell. T14 6T1 3 [4HM,66 pointsLet A = [a; az] be a 3 x 2 matrix and suppose thatAx =HH for X = [HJ and ~-H: for y=[] Then the following holds:=a1 + 2a2=a1+a=a1 +a2,H =a + Ja2 HH =3a1 + 2a2 + la; _ 27 laj + laz + la;=a, + Jaz.la1 + 2a2Submit AnswerSave Progress
41.66 poirts Suppose A = [az . ..a;] and B = [bt bi] are two 4 x 4 matrices so that AB = 12 What is Abz That is. what is A times the second column of B? Not enough information to tell. T1 4 6 T1 3 [4 HM,66 points Let A = [a; az] be a 3 x 2 matrix and suppose that Ax = HH for X = [HJ and ~-H: for y=[...
5 answers
An airplane was heading due east at 350 mph in still air and encountered 44 mph headwind blowing in the direction N 37 W. Determine Ihe resulting ground speed of the plane and its new bearing:The resulting ground speed of the plane Is (Round t0 twO decimal places as needed: )mph:The new bearing of the plane (Round t0 tWO decimal places as needed )
An airplane was heading due east at 350 mph in still air and encountered 44 mph headwind blowing in the direction N 37 W. Determine Ihe resulting ground speed of the plane and its new bearing: The resulting ground speed of the plane Is (Round t0 twO decimal places as needed: ) mph: The new bearing o...
5 answers
13 Problem. Prove the Schwartz inequality:la bl < |allbl:
13 Problem. Prove the Schwartz inequality: la bl < |allbl:...
3 answers
Ectors:Show by applying the formula that Q 1 m * Q invertible? If s0. what is the inverse of Q Show by applying the formula that Q is symmetric Keep in mind the rules with transpose: and (A_ B)T = B AB)T = 4T + B
ectors: Show by applying the formula that Q 1 m * Q invertible? If s0. what is the inverse of Q Show by applying the formula that Q is symmetric Keep in mind the rules with transpose: and (A_ B)T = B A B)T = 4T + B...
2 answers
(5 points) Prove directly that every convergent sequence is Cauchy: Do not just cite this as something from class, You need prove this. (Hint / Getting Started: To get started, assume {an} is convergent sequence. Since {an} converges, there exists L € R such that an 77 L. Show {an} must be Cauchy:)
(5 points) Prove directly that every convergent sequence is Cauchy: Do not just cite this as something from class, You need prove this. (Hint / Getting Started: To get started, assume {an} is convergent sequence. Since {an} converges, there exists L € R such that an 77 L. Show {an} must be Ca...
5 answers
What is the logical basis of the F-test - one-way random effects model? In your answer , specify the model and the null hypothesis of interest . (6]
What is the logical basis of the F-test - one-way random effects model? In your answer , specify the model and the null hypothesis of interest . (6]...
5 answers
6 In the following group, identify the amino acids with nonpolar side chains and those with basic side chains alanine, Ser; Arg; K, Leucine, and F
6 In the following group, identify the amino acids with nonpolar side chains and those with basic side chains alanine, Ser; Arg; K, Leucine, and F...
5 answers
In the 2022 General Elections to oversee the entire process a committee of 3 is appointed. If there are seven FNM members and six PLP members available to serve, how many committees are possible if there are at least 2 PLP members on the committee?Answer:
In the 2022 General Elections to oversee the entire process a committee of 3 is appointed. If there are seven FNM members and six PLP members available to serve, how many committees are possible if there are at least 2 PLP members on the committee? Answer:...
5 answers
Evaluate.J J (8x Y) dx dy( (8x +Y) dx dy =(Simplify your answer.)Evaluate the following iterated integral: 2 xcosy dy dx2xcosy dy dx=(Simplify your answer:)
Evaluate. J J (8x Y) dx dy ( (8x +Y) dx dy = (Simplify your answer.) Evaluate the following iterated integral: 2 xcosy dy dx 2 xcosy dy dx= (Simplify your answer:)...
5 answers
Li squares DETAILS 1 LARLINALGB 5.4.031.dedendent
li squares DETAILS 1 LARLINALGB 5.4.031. dedendent...
5 answers
A8y _ 0.262 = 501 8 0 Jy + 0.72 = 426 3 57* + 3 16y + 0 7z = 1420Thotu vaureeduitacetunbund cocely
a8y _ 0.262 = 501 8 0 Jy + 0.72 = 426 3 57* + 3 16y + 0 7z = 1420 Thotu vaure eduita cetun bund cocely...
5 answers
Polm 25. Il thar inor than ono answer eniar thom a5 comma separated Ilst Find all axact solullors on tha Intorval 0 < *8ain"(*) - 220(nulo (unples}
polm 25. Il thar inor than ono answer eniar thom a5 comma separated Ilst Find all axact solullors on tha Intorval 0 < * 8ain"(*) - 220 (nulo (unples}...
1 answers
Each given point is on the terminal side of an angle. Show that each of the given functions is the same for each point. $(0.3,0.1),(9,3),(33,11), \tan \theta$ and $\sec \theta$
Each given point is on the terminal side of an angle. Show that each of the given functions is the same for each point. $(0.3,0.1),(9,3),(33,11), \tan \theta$ and $\sec \theta$...
5 answers
100-80CHz_=CH-CHzCHa M.W, 56| 60 L 40M(56)20152025303540455055mle
100- 80 CHz_=CH-CHzCHa M.W, 56 | 60 L 40 M(56) 20 15 20 25 30 35 40 45 50 55 mle...
5 answers
Compound121110986 53HSP-01-943ppm
Compound 12 11 10 9 8 6 5 3 HSP-01-943 ppm...
5 answers
Express X = 4t-!_ y = 9t-2 in the form y = f(x) (Express numbers in exact form. Use symbolic notation and fractions where needed:)y(x)
Express X = 4t-!_ y = 9t-2 in the form y = f(x) (Express numbers in exact form. Use symbolic notation and fractions where needed:) y(x)...
4 answers
Question (3 points) #ListenWhat is the pH of a buffer solution that is 0.100 M in CH3COOH (acetic acid) and 0.100 M CH3COONa? (Ka 1.8 x 10*5)05.7403.7404.749.26Question 8 (3 polnts)W)LIstenThe half-life of first-order reaction is equal to constantwhere kis the rate0.693k0.693/kk/2k/0.693Question (3 points) @oLiscenWhich ion (accompanied by Na forms basic solution when dissolved in water?SO3OcrNO3
Question (3 points) #Listen What is the pH of a buffer solution that is 0.100 M in CH3COOH (acetic acid) and 0.100 M CH3COONa? (Ka 1.8 x 10*5) 05.74 03.74 04.74 9.26 Question 8 (3 polnts) W)LIsten The half-life of first-order reaction is equal to constant where kis the rate 0.693k 0.693/k k/2 k/0.6...

-- 0.019739--