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CtCalculale 4he integral : fff 26+yv, where K S 4he fangj e defined Ga X+y 42' 34 and 0 <z < 7 .write 4he Integral (n Cylindri cal Coordinatr8 Calculatle...

Question

CtCalculale 4he integral : fff 26+yv, where K S 4he fangj e defined Ga X+y 42' 34 and 0 <z < 7 .write 4he Integral (n Cylindri cal Coordinatr8 Calculatle the Integral

ct Calculale 4he integral : fff 26+yv, where K S 4he fangj e defined Ga X+y 42' 34 and 0 <z < 7 . write 4he Integral (n Cylindri cal Coordinatr 8 Calculatle the Integral



Answers

In Exercises $37-44,$ evaluate the integral. $$ \iint_{\mathcal{R}} e^{x} \sin y d A, \quad \mathcal{R}=[0,2] \times\left[0 . \frac{\mathrm{g}}{7}\right] $$

We want to integrate this function. But first, let's break the function into partial fractions. So let the function b equals to a over wine has be over y plus two. Because you were one ministry cross out the denominator here to the right set. You will get these. Yeah. Yeah, I'm gonna let Why goes to three. So that this time here and this time he goes to zero. So my weapons that we just believe he's got his start Beauties, My c u b minus. And I was fine. I'm gonna let my wine b minus two so that this part and this part goes to zero. Whether he says 38 my hand size can be so be is Latina will. Fine. And I'm gonna let my wife zero left. That will be minus truck. Right side will just be minus 68 because this goes to zero and this goes to zero. So my age is to Okay, so now I'm going to integrate my age too. My baby is 19/5, minus my over five. One minute. Do you mind? So when we integrate, we get to long. Why mark? Last 19/5 long model White. Last two, minus 9/5. Long off Worldwide Ministry meeting Upali meal to a local emir One. When you step in, you realize you're at least timing up. And also four is too long too. So cutting up you will get these. Yes. And we're gonna bring this tree up to here. So final and still be 19/5. One it over three. Now you notice that the model answer The textbook is 9/5 instead of 19/5. But the model answer is wrong. My answer is correct.

We want to integrate dysfunction, but we cannot integrate directly. So let's split it into partial fractions. Who left the function? Okay. Now notice that the denominator can be fact arised into this. So we're gonna let this be a over to express one plus B over X plus one across up this part, and you will get who it goes to a X plus one plus B to express one. You're gonna let Mexico's the minus one you will have on the left side is too. On Dhere, you goes to zero plus B minus to plus one, and you will get be at least minus two. Let as he goes to minus half. This part will go to zero. So we have two equals to eight minus half plus one zero. You get a is for So now we are going to integrate all over to express one minus two over X plus one. Yes. So this will be four one model to express one over to minus too long, off X plus one. And this is between upper and lower limits. 10 Okay, so that's something. So both terms have to, so I can affect the rest of the two. I'm going to stop in one first and then zero. So I will have 13 minus long to minus long. One minus long. One. Yeah. So these cancels anyway, So we have to long three over to as a final answer.

Or asked to evaluate this integral using the substitution u equals X minus seven. So to do that, let's find the you first. The U. Is going to be equal to the turret of of you, which is just one the ex. So now we can rewrite our interval using you and D u. So this is now equal to the interval of you to the power of three. And the ex is just you. So this is going to be equal to, um, one over four. You to the power of four. Classy. So now we're going to, um, plug. Are U equals X minus seven back into here, which means we get one over four times X minus seven. The trouble is for plus E.

We want to use the substitution formula in there, um, seven to help us evaluate this Integral here. Now, normally, whenever I see, like a drug electric function squared, I either think power reduction or I think with a gris, in this case, I don't think our production will help us. It'll since we'll end up live two times our angle. And then we have that co sign of data which I don't know how to really deal with what we have to sign of data and, like, co sign of data. So instead, let's try a fag, Chris. Maybe so or tangent path. A gris tells us we get seek it squared. Data minus one times co sign the details. Now, seeking is just one of her co sign. So if we multiply this by co sign, we should end up with just seeking fate of minus co sign data D sailor. Now I'm gonna go ahead and apply linearity of the interval to rewrite this as 02 pi third of secret data D theta minus the integral of 02 pi third of co sign Dada dee data. Now we know how to interact coastline But I have absolutely no idea what to do with that seeking. So let's go ahead and actually do that up here. So the integral of zero to pie Third of seeking theater D sailor Now the first step is going to seem pretty un intuitive and you have to get pretty creative toe. Think about this yourself and I'm going to go ahead and multiply the top in problem by Tangent X plus c connects well Tej it X while seeking X and then we also have to divide by tangent X plus seeking tax It may not seem obvious is all why we would want to do this. But if we let you equal to tangent X plus C connects now So you is eager to tagine x plus you get pecs Well, we get the differential. D'you is even too So the derivative of tangent ISS seeking Squared X and the derivative of seeking ISS seek int tangent do not be screwed all this over because I still need to write the d X on the end. Now we factor out that seek in we would end up with sequence data times tender necks plus C connects so everything we have in the new murder just becomes d'you and our denominator becomes you. So this is going to give us, Let's see, do you over you. And then let's figure out what our new bounds for this will be. So when fate A is equal to zero, remember, we're gonna plug everything into here so we get you is equal to so tangent of zero is zero. Seeking of zero is one. So are lower Balan becomes one, and then our upper bound is going to become so beta is equal to high third. So you is going to be so tangent of pie. Third is going to be so it's signed over. Co sign. So sign of pie. Third is Route 3/2 and then co sign of pie. Third is just 1/2. So I should just be hurt three. And then when we plug it into C can't Well, we already said that's going to be won over 1/2 because, see, it's just one of the coastline, so that should just be too. So actually, those counts out and we end up with Route three plus two for our new upper bound All right So let's go ahead and integrate this. Then you just plug that in down there. So that gives us the natural log of the absolute value of you evaluated from one to Route three plus two. And since both of our balance or positive, we could just go ahead and plug those in directly, right? So it's block old at all, and we can plug that in. So that's gonna be the natural log of the absolute value of you evaluated from one to route three plus to minus. So the anti derivative of co sign is just sign. There's gonna be sign of data evaluated. Trump 02 pi third. And then I'll go ahead and this up here so we'll have the natural log of Route three plus two and then minus the natural log of one, which is just gonna be zero. Then we subtract when we plug in Sign of Pie. Third, which we already said should be Route 3/2. And then when we plug in zero, Well, that's just gonna be zero. So we can simplify all this down to the natural log of the square root of three most to minus the square root of 3/2. This would be our final salute


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