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Compute the surface integral over the given oriented surface: F = (0,7,x2) hemisphere x? + y + 2 = 64 2 2 0outward-pointing normalAnswer:...

Question

Compute the surface integral over the given oriented surface: F = (0,7,x2) hemisphere x? + y + 2 = 64 2 2 0outward-pointing normalAnswer:

Compute the surface integral over the given oriented surface: F = (0,7,x2) hemisphere x? + y + 2 = 64 2 2 0 outward-pointing normal Answer:



Answers

Evaluate the surface integral $\iint_{s} \mathbf{F} \cdot d \mathbf{S}$ for the given vector field $\mathbf{F}$ and the oriented surface $S .$ In other words, find the flux of $\mathbf{F}$ across $S .$ For closed surfaces, use the positive (outward) orientation.
$$\begin{array}{l}{\mathbf{F}(x, y, z)=x \mathbf{i}-z \mathbf{j}+y \mathbf{k}} \\ {S \text { is the part of the sphere } x^{2}+y^{2}+z^{2}=4 \text { in the first }} \\ {\text { octant, with orientation toward the origin }}\end{array}$$

Giving that the S. It's equal to this. And you have sex giving us the square root of one minus X squared. Then taking the pasha directives on set with respect to X. And Y. And substituting you have the eggs to be equal to that. Therefore we can evaluate our integer. So the integral over the surface X. Of ex blood white bloods zero. The eggs is going to be equal. So in zika over tea. W. In Segre you have X. Plus Y. Z. It's like square roots of one minus X squared Divided by square roots of one minutes. S squared the X. The then why our D. Is still the projection of the surface on the xy plane. So hence we have D. How's the interval from -1-1 Then from 0 to 2? So we have fancy girl D. You have X. Plus white plus square roots of one minutes. S great. Divided by square roots of one -X. Squared the eggs. Dy it's equal to Thank you girl. From 0 to 2. From negative one 2 1. You have eggs divided by one. Mine. It's X. Great. Best square roots square roots of deaths. Then blood worry. We have square roots of one minus X squared. That's one the eggs the Y. So we face integrate with respect to eggs. So we have a pentagram from zero. To tune. This would give us a minute square roots of one minus X squared plus why? Sign in vase of S plus X. In the interval from negative 1 to 1 the Y. And this would give us so they stay. So this is equal to you have density cra from 0-2. You expand this and you have by why? By Y plus two. Dy Integrate with respect to Y. And we have high on two Y squared plus two Y. The interval from 0 to 2. And this would give us two bloods to buy soup. I bloods four. So we have to buy plus for as a final answer.

And this problem, we're going to be evaluating, uh, integral, the surface integral and more particular. Um, we can see how far back we can go to get to the formula, since we can't get the formula will rewrite it. So it's going to be the double integral of f of X y g of X y. So another function of X and Y and then that's gonna be times the square root of the partial derivative of Z or the partial derivative of deep of G, with respect to X spread class the same thing. The only difference is it'll be the partial derivative with respect to why that's gonna eat class one d a. So we'll copy and paste this because this will be the formula we're going off of. We see that since we have our hemisphere, um, were given initially that our f of X y Z is going to be X squared Z plus y squared Z. So have um, if we factor that out, though we'll have X squared. Using distributive property will have X squared. Plus why squared Times Z and we know Z based on the formula were given if we solve for it will be the square root of four minus X squared, minus Weisberg. And then we also want to take GDX or DZ DX. I mean, this is the partial derivatives. So what we end up getting once we simplify all of this, Um, just through squaring and then combining, like terms on simplifying, we will get four over four minus x squared, minus y squared E a, um and then this. Since we know that four the square to four minus x squared minus y squared is just, uh, z squared or the fact that these can cancel out right here. What will be left with is canceling these. So then we'll have the spirit of four, which we know is too. So we'll move that outside. So we've really simplified are integral just by following those steps. Um, by doing one thing at a time since X squared plus y squared equals four is the region. Um, it's gonna be inside that circle, so we'll want to switch to polar coordinates. That'll make it a lot easier. So it will end up. Having now is two times the integral from 0 to 2 pi. Since it's a circle, and then the radius is too. So we'll go from 0 to 2 x squared plus y squared equals R squared. And then we changed the A to our DRD data. So we'll make this our cube pr the data. Then we'll evaluate these any real separately. The D theta will just go right here, which will make this, um, two pi. So this together will be four pi and then, um, our cube, that'll go our to the fourth over four evaluating that we'll get times for. So our final answer will be 16 high. And as we see this is going to be the same answer that we calculated above.

We're trying to calculate thes surface integral of why DS And in this case since we have why in terms of Excellency Flight was expert plus C script, we can use the formula that thesis surface integral of f of x g of x z z multiplied by the um UAE. Partial with respect to X square, plus the white partial with respect to Z squared plus one d A. You can use this formula to chi chi d surface in the girl So we want our function to be in terms of X and Z are integral to be in terms of X NZ. So instead of writing, this is the integral why we're going to write. This is the Internal X squared plus Z square as shown above, Why equals X squared plus C squared and it's to be multiplied by the X Partial, which is two x So two X squared will be four x squared plus the Z partial, which will also be four c squared plus one d A. Now we're given that the bounds of from 0 to 1. However, as we have X squared and Z squared and four x squares and four Z squares. We can rewrite this in polder. Um, doing so we'll give the integral from 0 to 1 girl from 0 to 1, um, of X squared plus z squared. Same is above. However, this is going to translate from 010 to 2 pi and 0 to 2 for the DRD fate up of our squared multiplied by the square rid of one plus four r squared as X squared plus Z squared is equal to r squared. If we go to polar on, then this integral will be in terms of de are de Santa multiplied by our solving For this what are going to want to do a U sub for the square root as the four r squared is very hard to deal with, So setting you equal to one plus for our square. We then know that d u is equal to eight are and we want to get rid of this r squared right here with the use of So we're going to solve for our square using new equals one plus four are square and this will give that u minus 1/4 is equal to r squared. Now that we have all this information, we can rewrite the integral as 1/8. We want to pull out the eight here so we can set up a eight on the inside attached to the art. Ah, 1/8 times two pi times 1/4 of the integral from 1 to 17. 12 17 is coming from changing our bounds by plugging in zero and two into, um, And to our here of U minus one again by They were pulling out the 1/4 because you wanted to get rid of the four we played here, multiplied by the square root of you. Do you? And this will be equal. Teoh Pi over 16 spurt of 1 17 of you to the three halves, minus even 1/2. Do you, um this will be equal to once we solve this, cause this is an integral we can solve from here. This will be equal to pi. Over 16 multiplied by, um, you to the five have multiplied by 2/5. Minus one. Sorry. Not mean this one minus you to the three halves multiplied by 2/3. Um, from 17 to 1. And when we work all this out this will give a surface integral of pie multiplied by 3 91 Time skirt 17 plus one over 60. And this is thes, uh, area of our service, integral.

So we're looking to calculate the flux of our surface flux can be represented by the service in a girl of Flux Times DS. And in this case and Z is equal to or is he can be rewritten to be equal in terms of X and y. You could be using the formula that the sequence the integral of negative p times a g partial with respect to X minus que There's the G partial with respect toe wine, plus our d a. Now what do you mean by weaken right Z in terms of X and y well, since we're given the X squared plus y squared plus Z squared is equal to four we can solve for Z and write that Z is equal to the square root of four minus X squared minus y squared. So now we have a function in terms of, um, see being in terms of x and y so we can use the U formula buff and were given the F is equal to X I minus zj plus y que Now we need to come up with the X and y partials of F. And when we plugged this in this will be equal to the negative. Integral, I'll go over way. That's negative. In a second of negative X multiplied by negative X over Skerritt four minus X squared minus y squared. Um, minus negative C of negative. Why over square it four minus X squared minus y squared. Plus why, dear? So this is negative, by the way, because our equation is sloping downwards. And then this particular formula Onley where x one, we're going upwards so we can make it negative. And that will make you go from upwards downwards. So simplifying what we have here, we'll get that this is equal Teoh X squared over square it four minus X squared minus y squared d A. Um, we got rid of the Z, by the way by, um converting the sea which is equal to skirt four miles Expert minus y squared, um, week sought out. See, for through this right here. So now we just need to come up with the limits of integration. And we have expert in Life Square, which is a good representation, that we should be switching to polar. And it's doing so we'll give, um, in our value from zeros too, And the state of value from zero to pi over two and then converting. We'll have that. This is equal to zero to pi over two. Syria did too. Of our cubes coast sine squared data over the square root of four minus are square drd fada. And this will be equal Teoh negative zero to pi over two of coastline squared fada de theater I won't supplied by and a girlfriend is devoted to of our cubes over square it four minus r squared d r. So what we're doing here is, since we can write this as purely in terms of Fada and purely in terms of art were breaking apart and doing them separately, this prevents us from having to do any weird integration do the coastline. So solving this will give the first integral will be equal to 1/2 of theater plus 1/2 sign to theater from zero by over two, and this will be equal to pi over four. So now we have pi over four times the integral from 0 to 2 of our cubes over square at four to minus are square D r. Okay. And for this prom, we're gonna have to use a substitution. Um, setting four minus R squared, equal to you and negative to our d are equal to do you. And this will change our integration from 0 to 2, 20 to 4. Sorry, Ford is your So we'll have to We write. This has a negative 1/2. When a girl from his Drew 22 of our square I swear it's four minus r squared times negative two R d r. This will be equal to negative 1/2 in a girl from 4 to 0 a four minus you over square roots you do you We have to swap the order of integration by most when buying negatives this to become 1/2 you go from 0 to 4 of fourty to the negative 1/2 mine is t to the 1/2. Sorry. Do you and this of equal to 1/2 times eight teach the 1/2 minus 2/3 Tito the three house from 0 to 4 And we saw that outs that would be equal to 16. Interesting. This will be equal to 16/3. So to me, the flux we take the answer both in troubles won't fight together. This will be negative. Pi over four I want to buy by 16/3, which is equal to negative four pi over three and I would be the flux of our surface.


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