evaluate-the-given-double-integrals-int_0pi-6-int_pi-3y-sin-x-d-x-d-y-2

Question

Evaluate the given double integrals.
$$\int_{0}^{\pi / 6} \int_{\pi / 3}^{y} \sin x d x d y$$

Tyler Moulton · Accepted Answer

we want to evaluate the double integral given by the integral from zero to pi over six of the integral. From pi over three to Y. Our sin X. Dx Dy. So this question is found in our understanding of integration from the standpoint of Multivariate calculus are such let&#x27;s proceed to solve. So to solve we need to understand how to take an integral. What we&#x27;re going to do is you single variable integration techniques. We picked up in single variable calculus one by one where first we take the inner integral which in this case is the integral pi over three to Y of X. Dx. Once we evaluate that we plug it into the outer integral from zero to pi over 60 Y to solve. So we have in our double integral is equal to negative integral. Zero to pi over six. Cossacks from probably three dy dy or negative zero to pi over six integral coastline is happy. Why? Thus we can evaluate as follows, we have negative sign one minus one half white, from zero to pi over six, or negative one half minus pi over 12, which produces the final solution, pi minus 6/12.

Robert Daugherty · Answer

Section 82 We&#x27;re looking at Problem number 49 years. We have a definite interval power over six to power three of the co tension cube of X d. X. Best way to approach this. Let&#x27;s just do the anti derivative first, and then we&#x27;ll figure out the definite Inderal. As with many of these three, Key is going to be organizing this as powers of the trick function squared. So this is coach Ancient squared times the CO change it and then I know that the coach agent is CO c can squared minus one. So this becomes the integral of coast seeking squared of X minus one times a coach engine of X T X. Now I could break this into two into girls. So this is the coast seeking squared of X temps Ako tangent of X d X minus the integral of the co tended of X DX. Okay, so that allows me to break this into two of intervals and then I can go ahead and finished integration. So to integrate this, I know that you is going to be the co tainted of X. So let me just rate what I have on a different line here. So, um, we&#x27;re gonna have the integral of the coast seeking squared of X terms. A coat change it of X t X minus the integral of the coat chains out of X t X. And in this case, we&#x27;re gonna let you equal the code changes of X. That means that d&#x27;you is going to be minus the coast seeking squared of X dx. And so then what this gives us is this integral becomes, um, minus the integral you d you and that we have minus the integral let&#x27;s write code tangent as the co sign of X or with sign of x d x. And now we&#x27;re gonna let us have a substitution of but Z equal to the sign of X. And then D Z is equal to the co sign of X dx. So this becomes you. Integrate the first inter where you get minus You squared over two minus and this is the integral of won over Z, do you see? And so what this gives us is minus. You squared over two and when you integrate one over Z, that&#x27;s just the natural log. So that gives us minus natural log. Absolutely I z plus C Now let&#x27;s just go back to all of our substitution. Sze Yu is equal to the co change it affects. So this is minus co changes squared over too, minus the natural log Z. So what was Z Z was the sign of X. It&#x27;s a constant of integration. So that is our anti derivative. And we&#x27;re gonna need bad information to go back and figure out what the answer was to this definite integral. So we know that this answer is going to be minus co changing square of X over too, minus the natural log of the absolute value of sign of X. And we&#x27;re gonna evaluate this from high over six to pie or three. Okay, so it&#x27;s just a matter of substituting all of us in at things time. So what we&#x27;re going to get is we&#x27;re going to get my eyes 1/2 and then when you would substitute high over three. Ah, In this particular case, we&#x27;re going to get one and then minus natural log. And when you take the sign of that, you&#x27;re going to get square 23 over, too, and then minus We&#x27;re going to substitute in five or six and you get minus 1/2 on. Cho attended a pyre of six. It&#x27;s gonna give me three and then minus the natural log 1/2. And so now, homie, evaluate all of this. We&#x27;re going to get minus 16 minus the natural lawn square root of three, plus the natural log of to break that into two algorithms. And then we&#x27;re gonna have plus three halves plus natural log of one minus natural log of two. And what we see here, a natural log of two natural log on to that cancels natural log of one is zero. And so when you add this together three halves minus 16 your final answer is going to be 4/3 minus the natural log of the square root of three. So again, pretty long problem started out with finding the definite integral the Coastline Cube. What we needed was to write it. His co changes go change it squared, use both agree and identities, find the integration and once we have the final integration, were able to use this into finding this answer for the definite interval, which was 4/3 minus natural log of the square root of three

Ruirui Liu · Answer

So that&#x27;s Spokane. This unification, So won&#x27;t we calculated my ass? Our smee to zero to cycle, perhaps? Sure. The ex so we are can place. This is only, uh, careful. Oh, the, uh, immigration off this one. First psych. Thank you. Yeah. First. So we can&#x27;t have people. So who&#x27;s Count De Los? It&#x27;s like eggs. Yeah, DD opinion. It&#x27;s. And now we use in question my part too. Side arms, clubs. Thank you. Thanks. So we can&#x27;t know two I know. Yeah, actually to psych and yeah, my nights, you know, times, um, canyon ads. Where? Stipe. So these urges I my mom? Yes. And now you notice that these terms here, Actually, it&#x27;s a mess. The increasing one. Cap it some way through these term to laugh site. Now we can get Thank you Sake. So sure. Psych. Thanks and thanks. What? Bye. Thanks. Wish so these can be wise. Thanks, Max. X plus loan. All That&#x27;s Scott. Yes, I am. We Are you out of this English I hero. So we can&#x27;t. He was one we&#x27;d get here. It&#x27;s wrong. So they can&#x27;t Base wants to. I&#x27;m spiritually minus. Whoa, my scrubs

Robert Daugherty · Answer

section eight dot to problem number 39 The definite integral from negative pie or 3 to 0 of two times of seeking cubed of x d x. So many strategies I have Let&#x27;s just sort out How do we integrate seeking Cubed So how would I approach integrating the seeking cubed of x d x? Well, I&#x27;d probably say, Well, what would help if I would write? This is the seeking of X times, a seeking squared of X. Okay, and now I can do an integration by parts. So if I let you equal the seeking of acts, then d&#x27;you when I differentiate that is going to be seeking of X attention of X and then it Devi is equal to the seeking squared of X t X. Then I know that V is equal to the changing of X. So I have that the the integral of the seeking cubit of X d X is going to be equal to u V. So that&#x27;s seeking of X tangent of X minus the integral of the d u. So that&#x27;s going to be seeking of x 10 squared of x d x. Now I can rewrite tangent as seeking squared minus one. So this gives me the in a row of the seeking cute of X. The ex is equal to the seeking an extension of acts minus the integral seeking of X and then rewrite this a seeking squared of X minus one d x. So the interview of the seeking cubed of X d X is equal to seek an IMAX changing of X minus the integral of the seeking cubed of X plus the in a row, the seeking of eggs DX. Now you see this seeking cube in a grand occurring twice so I can move it from the right side of the equation to the left side of the equation, and I would get two of those, so I would have two. Integral seeking Cuban of X D. X is equal to seeking of exchanging of X minus three in a row. The secret of eggs said This gives me too integral of the seeking Cubed of X. D. X is equal to seeking of X tended of X minus when you integrate the secret. That&#x27;s the natural log of the change in X plus the secret of X so, plus a constant of integration. And so this will give me the integral of the seeking Cubed of X D. X is 1/2 of this expression so seeking of extended of X minus 1/2 natural log tension of X plus seeking of eggs plus constant of integration. So that is how I integrate the secret of acts. Now what I&#x27;m after, I&#x27;m after twice that. Okay, So much is on another page here. I know that two times the integral of the seeking cubed of x d x, it&#x27;s just gonna be two times all of that. So it&#x27;s just gonna be seeking of ex tangent of X minus natural log change of X plus secret of X plus a constant of integration. So now let&#x27;s go back and figure out this definite integral. So the definite integral is just simply going to be, um, seeking of X tangent of eggs plus natural log tangent of X plus, seeking of X, evaluated from negative pi over 3 to 0. And so, Alice, just do this evaluation. So So if I substitute zero in for everything, um, so at zero, the seeking zero is going to be one, but the tangent is zero is going to be zero plus and then I&#x27;m going to have the natural log change it is Zero seeking is 10 plus one into the natural log one which is gonna work out to be zero as well and then minus. So I have to substitute in a negative pirate three seeking a minus power three. That&#x27;s reciprocal of a co sign. It&#x27;s going to be a positive value. So that is going to be, too. The tangent of negative high over three that&#x27;s gonna give me a negative value. That&#x27;s negative. Square root of three plus the natural log and we&#x27;re going to have. So it&#x27;s gonna be, what, two minus the Route three. And let&#x27;s just see to minus root three is going to be a positive numbers. I can get rid of those absolute value signs. So this is going to be zero. That is gonna be zero. Your answer is gonna be to route three, um, and then minus natural log of two minus Route three. That&#x27;s your final answer to Route three, minus natural Log to Manager three. And the key here was to be able to take that seeking cube

Patrick Vaughan · Answer

Hello. So today we&#x27;re given a double integral and were asked to evaluate it. So the first thing that we do we&#x27;re getting a problem like this is that we have toe to determine the order of immigration. Well, just looking at both and girls were not given an explosive X or y variable it either one. So let&#x27;s look at the order of differentials in a d actually de wise that were given. So for this first integration, the one that&#x27;s closest to the variables, the first, uh, differential is with respect to X. So we&#x27;ll just start off with ex. You could start off with ingredient respect. Why doesn&#x27;t really matter in this case? Um, but, uh, just because we have the d X closer to the variables will just start with that and then we&#x27;ll integrate with respect. So why so the first immigration is going to have it so that we have a constant of six is retained and the integral, uh, what is signed? Wealthy, integral of co sign is signed and not just co sign but negative coastline. So we have our negative co sign of to act my three y, and then we have to remember that the, um also need to take this with respect to the integral of acts. So that, too, is going to be divided by that expression. Because when you take the integral of, um, co sign, it&#x27;s the derivative of co sign, which is I know you have signed times the derivative of what is inside the expression, and our two X is what&#x27;s of importance there. So that would be times to retain the value we divide by two. Okay, so this gives us, um, negative three co sign of two acts by three. Why evaluated from zero if I were to. So just plug in the street in, um, we can evaluate it such that negative three co sign of pie because it&#x27;s use Cancel minus three y and on this is subtracted from, um, negative three co sign. So that becomes positive. Uh, three co sign times. Uh, well, two X. We&#x27;re putting in zero now, so that becomes zero. So the expression is just three. Why? And then we could bring down the out of parts of the integration through the second immigration R D y, and then are integral from zero. However, to once more. Well, now we can actually take this expression and break it down into parts. So we have the integral from zero. However, two of mega story co sign of pi minus three y plus three in a girl of zero poverty to of co sign negative three y and then don&#x27;t forget are the wise B Why? Okay, so let&#x27;s take the integration. No. Well, what is the integration of co side? Well, that&#x27;s just fine. And a girl of Sinus co sites assign times Hi, minus three y. And then don&#x27;t forget that constant. And then we have the integral, uh, well of the inside or the first derivative. So we have that negative three. Why that that negative three. Sorry, that has to be accounted for. And then similarly, on the other side, we have the sign of negative three y divided by, uh, negative three once more. So this goes down Teoh one. So that&#x27;s sign of high minus three. Y, uh minus. I&#x27;m sorry about that. Minus 1/3. Sign of negative. Three wide. Sorry. I forgot the, um forgot Thea three there, so that&#x27;s just negative. Minus sign of negative three y, and this is evaluated from zero Teoh high over two for the entire expression. So putting in first the pi over two, we have signed, uh, high minus three pie over to minus. Fine. Uh, negative. Three pi over two for the last and expression. And then we have this minus, uh, sign of high minus the sign of zero as our two expression. Well, we know that the sign of the euro, Well, that zero and similarly, this I have signed up. I also zero. So the only real expression that we have a value is our sign of time on this trip I ever to and are a sign of negative three pi over two. So if you put this into the calculator, um, no, pretty a and radiance. So we have negative one minus one. Well, I can listen after of negative to when evaluating at this expression, and that&#x27;s everything we did to get there

Doruk Isik · Answer

in this problem. Restimulate internal crew seek on squared x e x between Syrian firing three. We own it to physical sense where we can take this outside internal and the interview. Beautiful sheep inspired ISS tensions, eggs. So we need to leave this function it for zero and I recently like that for entry start firing. We planned it to be two times screwed a three minus l tendency or zero. So we find the answer to be too spirited.

Problem

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

Problem 11 Easy Difficulty

Evaluate the given double integrals.
$$\int_{0}^{\pi / 6} \int_{\pi / 3}^{y} \sin x d x d y$$

Answer

$\frac{\pi-6}{12}$

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$\frac{\pi-6}{12}$

Basic Technical Mathematics with Calculus

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

Lily A.

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Joseph L.

Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus