Evaluate each iterated integral. (Many of these use results from Exercises 1-10 ).

$$\int_{0}^{1} \int_{3}^{6} x \sqrt{x^{2}+3 y} d x d y$$

$=\frac{2}{45}\left(39^{5 / 2}-12^{5 / 2}-7533\right)$

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you want to evaluate the iterated integral from 0 to 1 off the interval from 3 to 6 off x times square root of X square, plus three times why respect toe? I respect to X and then with respect to y so this double integral here we have to and intervals first, dangerous from 3 to 6 of X square root of X squared, plus three times why with respect to X. So it's the first integral we got to evaluate. And then we got to evaluate the result off that integrated between zero and one we respect why. So the first thing we got to do is to calculate the interval from 3 to 6 off x Times square root off X square, plus three times why we respect to eggs. So here, in this case, why act as a constant and that's a good thing to calculate this into. So when we have this type of integral, we get to think about the possibility of changing the variable because we have a square root, so it's very likely that the expression inside the route will be a new variable, and then we get to see if we have the derivative of that multiplying this square root. If it's not the case, then we will have to think about two grown metric substitution. Another thing. But in this case, we're going to see that that change your variable is going to work. So we're going to call you or in general, another letter different from X and y equal the expression inside the square root that is X square plus three times Why? Remember why it's a constant. So we get to find differential vax So the differential of you is equal to the derivative off this expression. Respect to X. Yeah, so we write it thinks way. Partial derivative respect to X off X square plus three y times differential of X. And so the sequel to and derivative of X, where respect to access to X plus and the differential of three times why respect to X zero. So we end up with differential of you equals mhm two x differential of X. Okay. And as we see here, this expression X differential of X appears inside the integral because we have X times differential of X over here, multiplying the square root off you and So we have all that we needed in order to that change of variable to work. So we have that the integral another thing we need and we're gonna do it before we we write the change of variable. We are going to evil. Wait. The limits of integration. That is when X Remember, in this interval here, X is a variable. So when X equal three, which is the lower limit of integration here, we're going to have you equals we put three over here in the change of variable, we're gonna have three square God. So we got we're gonna have nine plus and in this expression, there's no eggs, so it's gonna be stayed like like that three y. So when x equal three u equals nine plus three y and when x equal six, which is an open limits of integration. Here we have u equals, and now we substitute x X X equals six on the change of variables. So we have 36 square that is 36 plus three times why? So this is these are the new limits of integration. We gotta put in the new integral with the new variable. You So now we can say that the interval from three, 26 off x Times square root off X square plus three y differential of X is equal to first. We're gonna put this X next to differential vex in order to have a discretion just like we want. So it's injured from 36 off square root off X square, plus three times y times. Okay, times three. Sorry, ex differential of X. And that's not not really a necessary step. But it's better to understand that we are arranging this to have, visually all the elements we need. And so now we're gonna put the new variable. So the new lower limit when x equal three is now equals nine plus three wise. So we put it here. The new Berlin it is 36 plus three y and the expression inside the integral the integral is square Root off you because x square. But through wise you here and ex differential of X. Here is Theis expression here. So if we, uh, sold for the expression x differential of X, we get X differential of X equals differential of you over to. So here we put differential of you over to which is the equivalent expression to differential of X Different X differential of X. So this is our integral we have We have one health differential of you here we can put it outside of the integral one health intro from nine three y up to 36 plus three y off squares of you differential view And that's one half times. And now we can integrate the skirt off you easily. We used tables that you two the one half plus one over one health plus one evaluated between U equals 9 to 3 y and u equals 36 plus nine plus three y And so then we got when half times and one half plus one is three house. So we had we got two thirds because three house in the denominator is gonna b two thirds in the narrator times either you to the three house evaluated between nine plus three y up to 36 for three y. And that's equal to one third because thes to simplify. And then we have the upper limit my devolution of thes d expression at the upper limit minus the relation of the same expression at the lower limit. So we have 36 plus three y to the three house. Um, the three have minus nine blue three y to the three hubs. And that's the result off thes integral Over here. So now we get to integrate that we saw at the start of the exercise, we get to integrate between zero and one off the expression we obtained here. Respect to respect to y so the integral we're looking forward there is. This one is equal to the interval between zero and one off the result of thes interval here, which is thes expression here. So it's one third times 36 plus y to the three helps minus nine for the last three y to the three house and old dad is the value of the integral, the inner integral from 3 to 6 off x times square root of X, where for three y respect to x and that is now we're effect. Why? So now X is a constant, but we don't have x here. But if we had it, then we got to consider eggs as a constant. So we're gonna put it anyway. But we don't have that variable in this expression. In some cases, where you're gonna have that variable may be in this expression, and you're going to put that here. You are going to consider this. So now mhm this case at the end of the calculations, we cannot have a variable, because this, uh, definite internal so exchangeable off a function over a region in this case, a rectangle. So the result gotta be a number. So is not really logic that we will have a variable x at the end off the calculation. So now, um, right, we get one third as a constant on. Then we have the integral from 0 to 1 off, and now we can separate the difference of the integral. So enjoy all of 36 plus three y to the three helps respect why, that is the first term here, minus in jail from Syria to our nose nine plus three y to the three house respect to why all that multiplied by one third. So we got two intervals to calculate here. The first definite integral. We got to calculate its this one Okay? Yeah. And that for that we get to make a change of viable again. Here we put view equals I'm going to use. You can use in a little different from wise. So you go 3 36 but three y and then the differential of you is three times three which is a derivative of Baltes expression. Respect to why I remember why survivor, from now and now we know way Know that because we have differential. Why here? See negative off that wise is variable. So we get three which is the derivative off this expression here Respect to why differential of why and because we have Onley differential of why Inside the integral with applying the expression we are going to substitute We got Thio find the value of differential of why only And that is one third differential of you and this is discretion. We're going to substitute here instead of differential wise So we get the following now we're gonna put the limits again So we get Thio, put the limits here So when y equals zero which is the lower limit in this integral we have u equals 36 by replacing Syria and this expression here and when. Why? It was one which is the upper limit. We get u equals 39. So with this substitution here we get the result. We're gonna right here above We got the integral from 36 to 39 because these are the new Lower the limits. And then we have 36 for three y to the three house that is you, which is 36 for three. Health 36 plus three y sorry. Issue to the three house and differential of you we saw Here is one third time's differential of you. And we got a constant that get out of integral one. Third time Central from 36 239 of you to the three house differential of you. Now we're gonna break a little bit. This is going to go here, all right, down here. And, uh, this is equal to one third times, and now we have the integral of you to the three houses. Three house, directly using tables is you to the three house. Last one over three, half plus one, and that evaluated between 36 and 39. So it's one third times we got you to the, um five house over five house people waited between U equals 36 you equals 39. And that is a five half in the denominator get to fit in the numerator and then that constant with this other constant. Here the multiplication is too over 15. It's to over 15 times you to the five house evaluated between U equals 39 you equals you Go 36 you go third night and then this is equal to 2/5 15th off on in the upper limit, we get 39 to the five house minus 36 to the five half. This can be written us to over 15 times the 39 to this five half. Remember that three power 1 51 5 house corresponds to the square it off 36 to the field here. We don't do that because discouraged of 39 carry on into your number. So with left left sees. But in this case, we can calculate the spirit of services. Six, we got 39 to the five house minus six to the faith. And that's the results in America. Result off the integral. This one here. Okay, Now we got to calculate this other into your here. So we got to calculate that. So it's the second one is integral from 0 to 1 off nine plus three y to the three house differential of why. And again we do change variable here we're going to as before equals three. Sorry. Nine. Yes, 9 to 3 y. So differential of you is three. Differential of why and we solve for differential of you is one third differential of you. And now we do the evaluations of the limits of integration. So when Why equals zero u equals nine by replacing u equals you were here and the other limit, the upper limit is why it was one corresponds to you equals 12. And so with this we right, the new integral is integral from 9 to 12 off you to the three half differential off. Why is one third differential view? And so we could get one third time's interval from 9 to 12 off you to the three house differential of you. And so is we're going to calculate over here. Yeah. Okay, so it's one third times and this integral is you to the five half over five. Half that is three house plus one over 300 plus one. He will write it between U equals nine and U equals 12. So he's one third times thes constant here corresponds to to fifth and that times you to the five house evaluated between nine and 12. And that is to over 15 off. The upper limit is 12 you awaited at this. Russian is 12 to the five house minus nine to the five half. And that's equal to 2/15, 12 to 35 half minus. And this one can be written as discourage of nine to the fifth. Because it's worth of nineties and into your numbers. Very easy is three. So we get 12 to the firehouse, minus three to the fit. And so we have the results we needed to calculate. Now, these less integral here. So we can say that finally, the integral we are calculating what is one? Okay, that is okay. We look over here very important to remember we had here Control. We're calculating. We got one third times the difference of these two intervals between square brackets. So it's one third time's the first interval, minus the second one That is why we got to put here one third times Guru Square bracket off the first. Integral is this one here to over 15 times 39 to the firehouse, minus six to the 56 to the fifth. That is the first interval minus the second one, which is thes result here minus 2/15 times 12 to the five house, minus three to the fifth. And that is equal to We have a common factor to over 15. So it's one third time's to over 15 times 39. Now we can take out maybe the parenthesis. So we get 39 to the five house minus six to the fifth, minus 12 minus 12 to the five House. Plus, because we have here a negative before the parenthesis and negative inside the parentheses, we have plus mhm three to the fifth more. And if we simplify, the here is to over three times 15. It's 45 times 39 to the five house, minus 12 to the five half plus three to the fifth, minus six to the fifth. We do that calculation that's equal to negative 7533. But that's it. So we have this result. This is the interval we were looking for. This is and numeric values. So we have the integral off dysfunction of two variables that this surface here over every tangle the rectangle Siriwan times 36 So the integral of thes function over this rectangle is equal to this number here.

Universidad Central de Venezuela