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Problem 14 Easy Difficulty

Evaluate each iterated integral. (Many of these use results from Exercises $1-10 .$
$$
\int_{0}^{3} \int_{4}^{5} x \sqrt{x^{2}+3 y} d y d x
$$

Answer

$\frac{2}{45}\left(24^{5 / 2}-21^{5 / 2}-15^{5 / 2}+12^{5 / 2}\right)$

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{'transcript': "All right, So we have the double inner girl spirit of three for flies of backs Rent X squared plus three. Why? T? Why t x OK, and so the way is written. It's written. Teo do the integral of prospective wife. First of treating X is a constant. We need to find an anti derivatives of this immigrant with respect to why, but Okay, first of all, this X is Constance Will pull that out on, then we're just left with brood X squared. Plus three. Why? Well, wake unjust if we wanted to let you equal X squared plus three. Why? And if we differentiate with respect to why we have do you his three? Why do I just one third to you? Okay. So we can replace this with you in a girl here. So if we evaluate our limits at y ankles for X squared plus that of twelve, then X squared. Plus teeming. And we have what's left Just square root of you, Teo. Okay. There's a change. Variable. All right, so this just becomes in a girl, Teo three of X and then this becomes well okay. You to the three halves. Three halves the infirm. X squared plus twelve X square plus fifteen. Okay, Thanks. It's, uh, simple, but that down we have X times X squared. Plus fifteen, two, three house, I guess. Okay. So, uh, there's one thing Africa here. So this is one third to you. So we have a factor of one third. And then way also have this factor. So there's a one third beacon factor out. We have this factor of three halves, so that makes this bacterial here two nights. Okay. All right. And then So let's finish this guy off. Sir, minus X squared plus twelve to the three house. OK. And so if we let know, we're going to have to really do too. Substitutions, But they're going to be the same S o. Let's let you be, uh, x squared fifteen and then civil noticed that to you is to the ex d X. It's one half to you, but knows that if we let you be x squared plus twelve, we're going to get exactly the same thing. So I guess technically, we should call this like you prime. Okay, so we want to evaluate this integral. So this is what we're calling you. This is where we're going. You prime. But do you really is the same things to you? Prime, You play There's an X here. Ex ddx So we made the substitution of RDX out here we have our x t x that'LL get absorbed with the factor one half. Okay, so this one half comes out the two nights becomes one night. Ah, in a girl now, any change our limits that the limits were going to be different for you and you primed. So for you, we have fifteen, two, twenty four. This is gonna be, uh, see new to the three house, and then plus, this is going to be you prime. And then we need to change our limits of integration to you, Primal. That's just going to be twelve to twenty one. You pry into the three house to you prime kid. And so we have I am both of these anti drivers. They're going to be the same. We're going to end up dividing by five halves, so that will make this a to over forty five. And we have you to the five halves gyrated from fifteen. Twenty four. Plus you prime to the five halves evaluated from twelve to twenty one, So this should just speak to over forty five times twenty four to the five halves dinos. Fifteen of the five halves plus twenty one to the five has minus talk to the focus."}