evaluate-the-given-double-integrals-int_0pi-6-int_01-y-sin-x-d-y-d-x-2

Question

Evaluate the given double integrals.
$$\int_{0}^{\pi / 6} \int_{0}^{1} y \sin x d y d x$$

Tyler Moulton · Accepted Answer

we want to evaluate the double integral given as the integral from zero to pi over six of the integral from 01 of why Synnex Dy dx. In order to answer this question. When you have an understanding of different or rather integration in the perspective of signal very calculus we&#x27;re going to proceed in two steps to solve and step one. We use single variable integration techniques to solve the inner integral. That is the integral from 01 Why Synnex Dy once we evaluate this inner integral we use the result as the integration of our outer integral to solve. So let&#x27;s proceed. Alright. Double integral is equal to zero to pi over 601 Y seven X. Dy dx or the integral from zero to pi over six and y screwed over Tuesday next from 10 This is simply the integral from zero to pi over 61 half sine X. Dx. So since sign X has an anti driven negative coastline, we must have that are integral is equal to negative half coastline X from zero to pi over six. Or as a final solution, negative 3/4 plus 1/2 or one half.

Enkhzaya Enkhtaivan · Answer

So in this problem, we have internal from zero to, um internal from zero to pine for six. Skirt of one class sign. Thanks. Well, so this interim squared, remember, sentences is a bit tricky, but eh So basically, the intuition is simple, though whenever you see a square, the function you want to get rid of two square, so it means you want to right one per sign off X as they square something. So some nice work in function. And so there&#x27;s only a handful of formula we know about side effects. Which double angle formula are the half in a form us really? And if you play around with him, you&#x27;ll see that the half angle formula is going to be useful, right? And after this, well, you have this constant one here, and whenever you&#x27;re doing with trick functions and you see the number one, the first thing you should be that should be coming to your to your right is the famous sign squared plus co sign squared formula. So now we have this plus to co sign X over too Senate X over too. And then you would realize that this is the square of course, scientific, sir. Two signers, except for two. Therefore, our in through Integral immediately becomes something rather simple. So we were We retake the squirt. It&#x27;s It&#x27;s just simply Chris sign D x here, and we can do the antidote. There&#x27;s immediately. So this will be sign of X over, too, while two times as two times cursed sign off X over to and the valued at points zero six. So let&#x27;s do this real quick. Um, Howard has two times sign off. Hi. Over 12. Thank you. Two times co saying pi over 12 and that the lower and the 0.0 which will want to give Oh, negative. Minus minus two. It comes from the co sign of zero and the sign of Serious Just here. Now, technically, this is a miracle answer. So we can&#x27;t leave it here. But as it turns out, this proud and we couldn&#x27;t even simplify this furthers using some clever trick formula. So it&#x27;s trying to do that. So I&#x27;m going during the next speech here. So now this is basically equal to two times. No, I have it went to now, too. Course, And I have this whole thing, so That&#x27;s right. There&#x27;s two tires squared Subasic and want to plan by squirt of two and square toe the line of scrimmage to so they don&#x27;t have whenever screwed. Too sign of higher, too. And then one divide script, too. Cursed sign. All right, 12. Well, you could you could do this without this portion, Annie. And if you knew the numerical clothes from River Seine and personify were 12 What? That&#x27;s something most people haven&#x27;t memory. So we&#x27;re trying to figure out it this way. And now the kid kid trick is that to recognize them, this one over square of two is actually sign off. Pyro. For similarly, this is co sign off piles before. Now you see where I&#x27;m going with its right. So this will become two plus two times square of two times. Um so it&#x27;s one more time. Sign of high over for sign off. Pie over 12. My Askar, Sign off pie off the floor for a sign. Hi. 12. And this whole thing is actually equal to negative co Sign off. Hi. Over three. Why? By the sum of anguish Formula. So you, Khun Khun checked. So this will become to the whole answer becomes now to purse two squared to And of course I have pie or three. It&#x27;s Christian of 60 degree, which is just one house. So with the negative one house, this is equal to that. So finally, the answer is to bring a squirt of two.

Tanishq Gupta · Answer

The question is to wait to give a nitric pregnant internalised integration from zero immigration from 0 to 1. Why sign heads? Do I buy the X now moving towards the solution? You see if you&#x27;ll be knees given an immigration from you. Why? Why integration from? Uh, I would, but I think it has integration from zero cuckoo Sine X, the X in tow. Integration from 0 to 1 by Dubai. So So I&#x27;ve been this way. Get minus or sex? Why spend by limit from 0 to 1 solving this, they will get one minus pause upon toe which will be inquired. Sign square. But so this will be finalized for the one question. Thank you.

Kira Schwander · Answer

So we&#x27;re going to be working on evaluating this integral. So really to start off by integrating with respect, Izzie Because these moves for this in. So we&#x27;re going to have We&#x27;re gonna leave those y and X integral on the outside. So we&#x27;re gonna have you by an eccentric rose and then the derivative, the anti derivative of sign. Why, with respect to Z is going to be easy times Sign off why? That&#x27;s going to be evaluated from zero to sign of X. So we&#x27;re going to plug in sign of X and zero for each of these. For the upper and lower bounds of Z, it&#x27;s really going to have a sign of x times sign of way and zero times sign a boy so it&#x27;s gonna cancel it 20 on. We&#x27;re going to be evaluating with respect to X and with the specter. Why So now we need to find the anti derivative with respect to eggs. So we&#x27;re going to have the anti derivative from zero to pi of the anti derivative of sign of eggs. Oh, dear. That&#x27;s a wife. The anti derivative of sign of X is negative co sign of eggs that&#x27;s going to be multiplied by sine off. Why? And it&#x27;s going to be evaluated from zero to pi. So we&#x27;re going to plug in pie and zero for X. So we&#x27;re going to have negative co sign times pi up the coastline of pi times sign y minus Negative co sign of zero times Sign Why with Inspector y. So the negative co sign of pie? Well, the coastline of pies negative one. So that&#x27;s just going to be positive. And so we have sign of why minus the negative co sign of zero, the coastline of zero is one. So it&#x27;s going to be positive one. So we&#x27;re gonna have a sign of y minus the negative sign of way with the spectre. Why? So that&#x27;s going to be two times sign of why with respect away. So now we can evaluate this with respect to why so the anti derivative of sign of why, with respect to y is going to be negative to times co sign of why? And we&#x27;re going to be evaluating from zero to pi. So we&#x27;re gonna plug in pie and zero, So we&#x27;re gonna have a negative to Times co sign of pi minus negative to Times Co sign of zero. So it&#x27;s going to be equal to co sign of pie again is negative one. So there&#x27;s going to be two plus co sign of Zero is positive one. So two plus two equals four as the answer for this integral.

Megan Mcfarland · Answer

in this problem, you&#x27;re given these shown double integral and asked provided solution. Now we know that we are going to be solving this integral in terms of X first. And so we&#x27;re going to go ahead and do that so we can ignore the zero to pine now because that&#x27;s going to be the integral when we evaluated in terms of why. But for now, we&#x27;re going to focus on the 0 to 2, and we want to take the anti derivative of y side of X y in terms of X. When we do that, we&#x27;re gonna get negative co sign of X Y because a derivative of coastline is negative sign and so there&#x27;s no negative. So we need another negative balance that out. So they get negative coastline. And if we take the derivative of the inside in terms of X, we&#x27;re gonna get a y outfront, which we have. So that&#x27;s what we get. We&#x27;re going evaluating from 0 to 2 and then I really with the whole thing in terms of why. And so when we do this, we get negative co sign, uh, we&#x27;re gonna plug the zero in the zero in the two in four X because we evaluated in terms of experts. So too, why you have close enough to why, minus negative co sign of zero. All of that in the integral from zero to pi and then t why? So when we evaluate this, we get the integral from zero to pi of negative co sign of two. Why co sign of zero is one and the two negatives consulted from a positive. So we get negative to co sign of I plus one and again D y. Now we&#x27;re going to take the anti derivative of this. And when we do this, we get negative. Sign B negative. 1/2 sign of two. Why plus why? Evaluated from zero, it&#x27;s a pie. And when we do this, we get negative. One house sign of two pi plus pi minus negative. One house sign of zero plus zero sign of zero and two pi zero. Syria was zero and we get pine and we have found our answer to our double integral

Carlos Pinilla · Answer

all right. You to go from minus stool? The three of why Zainal? See, e x. Why Z under here for Why? Yeah, That is from zero up to one on for C from zero to. By where? Six. If you if very stuff into go. Oh, exit. Just really X the x General of the X Just that So the excellent three minus two. So that is gonna be three minus my guest. Still so then becomes a place of three plus through that five. So the people too. Five times inter forms Europe by over six from job one off. Why sign of Z Do Why do you see me Will Now we&#x27;re integrating. Why do you want that? Has a direct Why word house for that between 10 with just we have minus you. Well, it would be just 1/2. So you have. But these she legal five clubs Inter from zero up to by over six. Well, sign we&#x27;ll see is you on. And, uh, well, no general sign, you seem drop in trouble. Size minus. Go inside. So we&#x27;re all waiting. Meeting five or six zero. She will be people minus consign off by over six. We&#x27;re on minus minus, cause I know zero. This one I&#x27;m for Why are seeks of the the unit circle where you have Ah, well, by over six is like, this is by set up by our six would be sort of like he&#x27;s much by over six. Were all these business should be one house. What is this sign? No, this part is gonna be Oh, sign strangled by over six. So in very simple truth. Hey, you one have squared, which is this sign? Plus these distances squared has to recall one. How do you square? We&#x27;re on my phone, Jill 3/4 So that whatever six is the square root of that would be squared off Three for two square the he&#x27;s number from these one. So if it is minor, that would be one minus square with three over to under the old ladies that integral. So you with me only that&#x27;s what would be my house. One minus x squared with so free divided by two. These should be than usual. She won&#x27;t do. He&#x27;s number

Problem

Evaluate the given double integrals. $$\int_{0}^{…

Problem 16 Easy Difficulty

Evaluate the given double integrals.
$$\int_{0}^{\pi / 6} \int_{0}^{1} y \sin x d y d x$$

Answer

$\frac{\sqrt{3}}{4}-\frac{1}{2}$

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$\frac{\sqrt{3}}{4}-\frac{1}{2}$

Basic Technical Mathematics with Calculus

Lily A.

Anna Marie V.

Kayleah T.

Michael J.

Vectors Intro

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Allyn J. Washington, Richard S. EvansBasic Technical Mathematics with Calculus

Lily A.

Anna Marie V.

Kayleah T.

Michael J.

Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus