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Find each double integral over the rectangular region $R$ with the given boundaries.

$$\iint_{K} \frac{y}{\sqrt{2 x+5 y^{2}}} d x d y ; \quad 0 \leq x \leq 2,1 \leq y \leq 3$$

$=\frac{1}{15}(316-130 \sqrt{5})$

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Okay, another double integral of erecting. So go and put in the exxonm Our limits Exes from zero two And why is from one two three? The function is why over square it's two X plus five Why squared and let's see So I put why on the inside So it'Ll be d y the x and hear this function Yeah, it looks to be continuously defensible in this rectangle, so it's not gonna matter. We can switch the order of of integration. So those Dubai first and this looks like a U substitution problem. So if I let you be two x plus five y squared to you too with respect of why Saudi ten Why b y And so why do you know why that we have on top here is one tenth, you know. So this is okay outers the same dear to two. And then what do we get if we play in one and three for Why? For you, we get see to X plus five and then to ACS plus Ah, see five times nine. Forty five. And then we have what's left. While we have this factor of one tenth from this institution the intern grand just becomes you to the minus one half. Then we have our DEA and T X Okay, So anti derivative of you to the one half needed minds. What happens is you the one half divided by one half, which is the same thing is multiplied by two. So if we factor that out, we'LL be left with one fifth out here Here too. And then we have see to X plus forty five one half minus two X plus five to the one half And now these anti drivers they're going to be very similar. So we're gonna have tio We'LL see who add up to three house we'LL divide by three halves which will be the same thing is multiplied by two thirds So get a factor of two thirds for me One fifth time's two thirds But then with the two X, we're also gonna have to divide by two since times one half and then we'LL have two X plus forty five two three house Linus to X plus five to the three house evaluated from zero to two. Okay, so what is this? Well, this becomes one fifteenth And what do we get when we evaluates all of this. Okay, so when we put into here, we get forty nine, which is square it. That's going to be okay. So seven cubes of forty nine times seven. What's going to give us three. Forty three? Because that seventy three house minus okay, plug into will get nine nine to the three halves is twenty seven b minus twenty seven and then pull us. My weight is zero. Forty five to the three halves and then the value is your AA minus wired to the three hours. What is this? One fifteenth, three forty three minus twenty seven, three, three, sixteen. And let's see, we'Ll just leave this forty five to three halves minus time to the three of us.

Georgia Southern University