Evaluate each iterated integral. (Many of these u…

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Georgia Southern University
Problem 14

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

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

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Video Transcript

Okay, so here we have the stubble Inner girl zero to three for two, five and x square it x squared plus three. Why? Right now it's the white the X s. Oh, it doesn't matter whether we do y Rex first. I'm actually going to change the quarter first thing, and that's more or less a personal reference of what I see going on. It would work out just fine if you do not do tax first. But, um, I see this ex hanging out here. And if we do the acceptable first it'LL actually get rid of that. So what I see is this x squared plus three y into the square root. And when I differentiate, I'm going to get something like X t X. So that's why I'm going to do X first. So if I let you be X squared, it's thirty. Why two years? Two x, dxe or X? The X is one half, do you? So we have an incredible forty five change our limits. So free plug in zero for X will get three why and then nine plus three. Why? And now we just have you to the one half to you see? Why? Can't breathe Specter on half for a substitution. Okay? And so this is just going to give us you two three halves over three house. So if I factor out that won over three House one half divided by three houses. One third. So we'll be left with factor one third outside. Then we have just the the end points evaluated, uh, into this function. You did the three halves Now since nine. Plus three. Why two three hands Arness three. Why two three house. Why now? If you look at these functions where you take anti derivative, it's going to involve dividing by five house and use the power rule and dividing by three when we use the change room. So this is one third finding my to have says the same. Fine house says the same thing is multiplying by two fists in the mirror. Ross a dividing by three That said And then we just have nine plus three. Why? To the five halves minus three. Why five have and evaluate from for five. So this fraction out front just gives us too over forty five and then we just want to evaluate at these functions at five so nine plus three times five cast with teens and twenty four the fight has minus fifteen. The five has and now evaluated for We'LL have minus twenty one to the five halves and then minus minus plus twelve to fight house.