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Problem 24 Hard Difficulty

Find each double integral over the rectangular region $R$ with the given boundaries.
$$\iint x^{2} \sqrt{x^{3}+2 y} d x d y ; \quad 0 \leq x \leq 2,0 \leq y \leq 3$$


$\frac{2}{45}\left(14^{5 / 2}-6^{5 / 2}-8^{5 / 2}\right)$


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Video Transcript

okay who were evaluating this double Integral of strict angle. And I'm to go ahead and put in the limits. So why's coming from zero three and X is going from zero to two, and then the function is X squared. Read, execute Plus tio I area of D x d. Why? Okay, so we're going to take the ex and her first. If we look here, we have excuses to Why the derivative of that? With respect, Axl gets a factor back squares, we'LL let you be execute plus two. Why do you three x squared X or, in other words, X squared X is one third Holmes Nichols one third to you. So we have a factor of one third here. Running substitution. We still have the girl from zero to three of why. But now let's change. Are you limits? So an ex is zero. We have to Why? And then we have eight was too high. And then what do we have left here? You to the lower half. You still have d y really? So since one third seared three. And then here we have you to the three halves by over three halves. Wade from two. Why two eight plus two. Why? It's salty wife. So this three house we can divide one third by three. Housing too. Thanks. And we have OK, two, three. This is a plus two I to the treehouse Linus two. Why? To the three house do you want? Hey, come up here. So now we just need an anti derivative and evaluated at from zero to three. So apes to y two three house and too wide and three hives are basically going to have the same and derivative. We're going to end up with the factor of on we're going to buy buy factor find has by the foul power raised the power to the five house and then divide by two. So another return and divide by five houses, same things multiplied by two fists. But then we have two multiplied by a factor of one half for the two. Why? And then we have eight plus Teo. Why? To the five halves minus? Why? To the five house. And then we're evaluating that from zero two three. So this out here gives us until to over forty five. And now we just need to evaluate Okay, So when we plug in three, the birth of these will get. So she apes to us fourteen to the five hands and then minus. So be six to the firehouse. And now when I plug in zero, I'll get plus eggs to the fire house and then minus sirrah.