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7_ Change the order of integration of 2x 2-2x2 L' K" % f(w,y,2) dz dy dx to dx dy dz...

Question

7_ Change the order of integration of 2x 2-2x2 L' K" % f(w,y,2) dz dy dx to dx dy dz

7_ Change the order of integration of 2x 2-2x2 L' K" % f(w,y,2) dz dy dx to dx dy dz



Answers

Change the order of integration. $$\int_{0}^{2} \int_{2 y}^{4} f(x, y) d x d y$$

In this problem were given the shown double integral and asked to switch the order of integration. Now, right now, we're first integrating with respect to why someone like a D y and the this inside interval here. Now, according to this, first we have to sketch our region are now according to this the occur the Y values we enter the region that why's it with the two X and we enter the read exit The region I Y is equal to two. So first it's Kraft y is equal to two x, so that survives of to and a run of one. So we have a Y intercept of zero. The next intercept of zero and so up to over one would bring us here and then here on. And if we go, the other direction will find that as well. And so this is our line. And so that's where we enter the region and they exit the region that y is equal to two, which is right here. And so as you can see, we have found what are why bounds R and R expounds are from 0 to 1, and so X is equal to zero right here. I'll just do it a different color. Actually, I'll make it blue. This is where actually equal to zero and X is equal to one right here. So we have found our expounds and so we know so again we why you enter the region her y values that why's it with the two actions right here And you exit a wise able to to So we're integrating in this direction. And this region right here is our region are And so again this line is why is equal to two X This here is why is equal to two and we have excess equal to zero and X is equal to one Now we want to switch the order of integration, which means we want some f of x y and then integrating with respect to x first not to do this, we have to find our bounds. If we were to integrate this way, where do we enter the region of our meaning? Right here. What are the X values there now? Right there. X is always equal to zero no matter where you enter along that line among the region and so we're first going to put zero. And then where do we leave? We leave along this line here. That line was, why is he going to two X? But now we want that with X, I itself. And so when we do that, we're gonna get why over two is equal to X. And so we enter a X equal to zero and lead X equal to y over two. Now, for a second variable, we just want the story number and the ending numbers. So when we answer X equals zero, excuse me. When we just look our though its Y value and our biggest Y value, why is equal to zero is our lowest value. And up here, as you can see, uh, why is equal to two that is going to be our greatest y value, And so why goes from zero to? And we have officially and successfully switched the order of integration

In this problem, we have to change the order off the integration. Now we're given that double integral from one toe on from zero to L N Y f or X Y d X de vie. The first will draw the graph. Now here. This one here is by exist This one here is X axis and this region here is the region off integration now double integral one toe and zero toe L N Y F or X Y B X de vie is equals toe double integral from zero to Ellen Toa and E to the power eggs toe to f off X y de vie d a. So that's a solution.

So in this problem, we're gonna be switching the order of integration, which is a common technique used to make integration simpler on, but it's very helpful to use, Um, and it's even more helpful to be able to recognize when we need to use it. So what we'll have in this case is, um, the graph of X equals Y over to yeah, and then actually equals one. Okay, so we realized that with this, it would be easier to switch the order of integration because how we can view this instead is since we're looking at this region right here, we see that why is equal to two X, and it goes from 0 to 2 x, and then X goes from 0 to 1, and that's much simpler to integrate. So what will have instead now is going to be the integral 0 to 1. This is the ex integral of the integral 0 to 2 x. This is the white integral of Why co sign this all stays the same. Execute minus one D Y. D. X. So we didn't change anything In terms of the calculation, we change the way in which reviewed it and that makes it much simpler to calculate because now we can calculate this portion of the graph, which we see will ultimately end up being okay. Two X squared co sign of X cubed minus one. And then, um, we can use substitution method X cubed minus one. Um, like that equal. You do substitution. And what we end up getting is that this whole thing will equal to third time's the sign of one. So with that, we see we get the same answer which verifies our original calculation on but just shows us how it's easier to do. Um, sometimes it's easier to switch the order of integration, which is important thing to recognize.

Okay. So as we can see, this is, um, indefinite inch girl first after authorization. We saw this exponent exponent. Looks not very nice. So what do we do? Here is, uh, we first apply change of variables. We introduce another variable X and a set X equals to 0.2 times. Why? Just this exponent exponent here. All right? Uh huh. After we do this, we saw why, in terms of X, so why were equal to five effects? Okay. And since this integral involved this dy So we do differential on both sides. So we know dy were echoes too. Five D X. All right, so after we we represent why and d y in terms of X, uh, we plug it in all right, to our formula. Yeah. Uh huh. The integral echoes, too. Five x times e to the power X times five times, DX. Um Then they pull out those constant. This will give us, uh, 25 integral x times e to the power X, the X and this part. We could use a integration by parts. So we first step is we put this exponential function inside the differential, so this equals to 25. Yeah. Integral E x, the u to the Power X right, So that then we use integration by parts that will give us 25 times x times due to the power X minus. We change those, too. I will give us, uh, You change those too, So this will become into the power x integral. All right, then. The next step is to just, uh we find the anti derivative for E to the Power X, which is itself. So let's so so the next page. All right, so the result will be 25 times x times into the power X minus into the Perec's. All right. Oh. Then we recover our original variable. Why? What we do remember our x echo 20 going to why? Yeah. So this, uh, substitute X into the formula will get This is an indefinite integral echoes to 25 times 0.2. Why? Minus squad times into the power 0.2. Why? Yeah, and this will be our answer


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