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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given this integral 01 and then the aero from 3 to 6 of the function X times the square root of X squared plus three wine dx dy y. We want to find a numerical answer for this double integral. Okay, so let's start off by evaluating this inside Integral first. So we have the integral from 3 to 6 of the function x times X where the square root of X squared plus three y dx Um So what we're gonna dio is we're going to use U substitution. So we're gonna let u equal basically X squared plus three y And then since we're integrating respect to DX, um, basically, this use substitution will be for X, So we're gonna be driving you or the left side by you So the left side will be to you, and then we're gonna derive the right side by X. So when we drive the right side by X, we consider every other variable to be constant. So we have to acts, obviously. But then this three, why could be considered as a constant? So this just just result in zero Andhra member to multiply by d accept and so And then what I like to do is I want this to match up with what I have in our integral. So I just divide to buy, but on both sides and I get that x d x is equal to do you over to Okay, so, plugging all this back in we have that, um, are new. Integral is, um basically, I should so, too with some stew x dx with do you over to so 1/2 and then do you I'm just taking out the constant right here. Just make it easier. And then this square root of X squared plus three, why should be just to live square of you now looking at the limits. Um, since we're deriving respect to you now we have to find the U Uh, basically, you limits instead. So we're gonna plug in our ex limits and sue here in order to get Are you limits? So playing three for X? Um, we have that you is equal to nine plus three y and then plaguing six and two here. We should get 36 plus three Y Okay, So this is our new integral. Um, Now, evaluating this can get a little messy, but bear with me. Um, we have 1/2 times we're gonna use the reverse spiral again. So we're gonna add one to the exponents. So 1/2 plus one is three house, and then divide by this new exponents should just result in 2/3. Um, and then we want to evaluate this over our limits, which could be pretty messy. When you're dealing with double into girls. Let me write that. Okay. Um okay, so now playing this in, you get 1/2 times 2/3 36 plus three y 3/2 and then minus now, playing in the bottom of it, we had 2/3 times nine plus three y all to the three house power. Um, Now, what I'm gonna do is just factor out this 2/3 so we should get 1/3 on the outside, and then everything else should remain the same. So we have 36 plus three y quantity to the three halves that minus nine plus three y quantity to the three halves. Um, now, this is just the inside internal. Now we have to pluck this inside integral result into the outside. Integral So we have been a girl from 01 of 1/3 just going to take this constant out right now of 36 plus three y 22 the three halfs minus nine plus three y quantity to three halves de y. Now you may be thinking that is little difficult to integrate this, but you'll notice that basically, we have a linear function inside of the power function or inside the power function. So I'm going to use a little trick that comes from you Substitution. But it's just a lot like easier to do and just makes it a whole lot faster. So we have 1/3 times the anti derivative. So we're gonna treat this inside linear function as if it were just like a Y, for example. So, like why? To the three halves. And if we were trying to anti Dreyfus, we would get we would add once of the exponents and then divide by that new exponents. So we have 2/5 okay, and then plus C. But we're doing it definitely. Girls said that doesn't really matter. Um, so applying this concept to our problem, we have 36 plus three y Also, the five have his power on 2/5 because we're defined by five house. So it just built location by the reciprocal, which is 2/5. Then we wanted to buy this entire thing by our coefficient linear coefficient, which is just three. So the next one would also use the same tactic because this is a linear function here. So we have minus nine plus three y toothy five house power no times to fifth on the divide. All this by our linear coefficient, which is just three we want Evaluate this from 0 to 1. So the reason, or basically let me just verify that verify for you that, um, this trick works. So if we look at trying to derive this entire thing here, Well, what we would first do is use the power roll so we would bring this five halves down and this would cancel with our two fists. Subtract that new exponent by three halves or by 1/2 by Sorry by one. And you get three house. And then, um or actually, let me just write this aside. So this is easier. Uh, So if we were trying to take the derivative of this. We were trying to derive this. Um, I would first use the power rule. So we have two times five or 2/5 times 36 plus three. Y um, Now we take this five house power, and it should cancel. If our 2/5 said those cancel, then we have three house here. This is the by by three. And then we have to use the train rule because we have a function inside of another function. So because the inside is a linear function, it's very easy to take the dribble of that, and it should just be three. So we multiply this by three. It cancel it this and you would be left with the original thing. Okay, Let me just racist. Okay. So verifying that, um, we can move on to after evaluating all this. So we have 1/3 times plugging in one. We should get two times 39 to the White House. Bye bye. Five. And then divide this by three than minus two times nine plus three should be 12. The five has five by five fold is survived by three. Then we want to track all this by what we plug in or the value when we played in zero. So should be two times 36 plus three times zero is 36. It's a five hives. Bye bye. Five to buy by three. That minus two times. Um, nine plus three times zero is 9 to £5 by by five by three. All right, so once we have this, um, well, we know that, um let me just try to simplify this a lot more before we move on. So, basically, all these, um, components are being divided by three. So I'm just gonna pull out the 1/3 first. So we have 1/9 on the outside, then we have 2/5 left on each component. So let me write that minus two times 12 5 halves over five. And then since we're subtracting this country, be negative, and this should be positive. So minus two fists, 36 5 hives and then plus two times nine to If I have over five now, I'm gonna factor out the two fists, so we should get to 45th on the outside because two times one is two and 19 5 45 So now the in shy inside should look a lot nicer. So 39 um, to the five halves, minus 12 to 5 halves and then minus 36 5 halves plus 9 to 5 halves. Um, make sure. Okay, then, once we're at this part, um, we're gonna keep the 1st 2 components in there in simplified form. Because if you tried some file result a dustman, um, this part So applying the to to both of these numbers on the inside, we're basically just square rooting them. So if I were to rewrite, thes would be basically the square root of 36 and then all of this fifth power, and then this would be square of nine all to the fifth Power. And we know the square root of both of those numbers so we can simplify that a little bit. The square 36 that's 66 the fifth, and then we have plus three two on this monstrous number is our final answer.

Rutgers, The State University of New Jersey