Write an equivalent iterated integral with the order of integration reversed_ J" ( f(w,y) dx dy y/2...

Evaluate the iterated integral.

$ \displaystyle \int_0^1 \int_y^{2y} \int_0^{x + y} 6xy\ dz dx dy $

So for this problem, we first want to sketch the region on Ben. We do this in order to change the order of integration. So, um, in this case, a region is going to look like our region is going to look like X equals three y. And then, um, with this, we have that this is where zero is less than or equal to y, which is less than or equal to one. So this is the region that we're focusing on. Um, And now with this in mind, we want to change the order of integration. So we see that why is equal to X over three? Um, so with that in mind, what we can dio is why goes all the way up to x over three and starts at zero, and then zero goes from or X goes from 0 to 3 so we can integrate from 0 to 3. Um, integrate zero to x over three. That's going to be e to the X squared B y the X. Because we switch the order of integration. Um, and then after doing that, we can take the derivative of this right here, which we find to be X X over three times e to the X squared. And then, um, we use substitution method and ultimately, we find that this is equal to eat of the ninth, minus one over six. The whole thing's over six and we end up getting the answer we're looking for, which is, um, this approximately or exactly this. And that shows us how changing the art of integration makes into rules much easier to calculate.

Right here. Is they iterated? Integral. So first we're integrating with respect to X And then with respects to y So we got an integral from zero to two blue integrate with their specs. Toe acts were keeping the white constants or than X cubed over three, evaluated from zero toe y squared. And then we're going to intubate with their specs to walk. So I'm gonna pull the one third out here, and when we plug in, why squared? We're going to get why, to the six only plugging zero, we get zero d y. So we write down What are into girl is now it is one third stooping to go from zero to of why to the seventh T Why? Okay, we can integrate this very, very easily. We know how to do this. From Cal Kwan zero to two no. One over twenty four times two to the eighth, minus zero to the AIDS, which is just zero. So our answer is two to the eighth over twenty four, And, uh, we can just use your calculator to figure out the answer from here, Okay, But this is a totally acceptable answer in my my book, and we're done

All right, here's a meteor and iterated interval, and we're first going to integrate with respects too wide. So that means everything with an accident, it's going to be sit considered a constant orjust our exes are gonna be considered a constants. So this first term here is X over. Why? So the X is a constant. So what we really have is X times one over. Why? An interview of one over Why, with their specs to wise, absolute value of Ellen. Okay, it's our first anti derivative. And then we've got y squared over to why squared over two. That's the integral of why and then over X. And I'm just going to write that as well. Times one over X Okay to evaluate that from one to two d. X So we'LL use FTC first on this first interval that we've figured out. So only plug in two for Ellen Ellen of two. Plus we're yet four over two times one over x minus X times ln one plus one over two times one over x d X men, sir. Let's remember, the Ellen of one is just zero and that we've got some, um, simplification that we can we can do here. We've got into row from one to four of x times. Ellen too. Plus two over X minus one over to X. And we can keep doing some simplification, like with Ellen two ext. Time zone of two plus three over too. Times one over X gx. It's This is a single variable in terms of our single interval in terms of one variable the way that we're going to solve this, we've got Ellen two times X squared over two. So that is the integral That's Auntie Drew. First term plus three over too. Ellen, have value of X from one to four. They were in use off to sea again. Here, your own of two times sixteen over to three. Over too. Dylan. Four niceness. Helen of two terms. One over two, three over, too. When of one. Remember, Ellen of one is zero. So oops. Soon equals. So what do we have? You got Ellen of two times fifteen over to plus three over too. Dylan. Good for you. So if we really look at this, there's some simplifications, some nifty nifty tricks that we khun D'Oh, I got fifteen over too. Times Ellen of two plus three over too times ln of two squared. I do remember from your algebra class we can bring this to to the front multiply fifteen over too. Own up to plus three tellin of two and no, we can. After just a little bit of outbreak manipulation, we can rewrite our answers. Twenty one over two Ellen to get her done.

So here is a iterated into girl and we've got zero toe on integral from zero to one of integral sort of one of the quantity X plus one. Why squared dx t y. Okay, So the way that I like to do these into girls Ah, instead of having to do U substitution, let's actually foil, listen to grand out. Okay, so when we foiled us into grand out, we're gonna get to X squared. I'm sorry. We're going to x squared. Plus two x y plus y squared, Jax, do y All right. So now this first integral here will underline it. They're all circle and red is the inner into girls I like to call it and we're going to integrate this with respects to X. So everything that does not have an accident is not considered a variable, but a constant. So this first one here x squared integrates Justus Usual. Plus this two x Why is why is it considered it constant? Time's integral of two acts is X squared. And over here we got Plus why square in which is a constant and integrated constantly yet y squared x. We're going to evaluate this guy from Zurich of one, and then we're going to take the the integral of that again. Okay, so when we plug in one, we're going to get one third. That's why. Plus y squared. And when we plug in zero, we're going to get zero for each one of these three terms, right? Each one of these three terms value it at X equals zero is going to be zero, right? We're just using FTC here. The fundamental fear of calculus. All right. And we've got this d wild over here. All right, so now this is a single into go in terms of one variable, and we know how to solve these. Right? So we're going to integrate this swinging at one third. Why plus y squared over to, plus why, Oops. Plus, uh, why cubed over three. And we're going to intubate that from zero to one. And the same thing happens as it has happened in the previous step. When you evaluate this into girl or this expression at one, we're going to get one third plus one half plus one third. And when we've hoops, when we evaluate it at zero, it's going to be zero. So our answer here is gonna be two thirds plus one half, and that's going to be right for six plus three six. So our answer is seven sixth and that's it.