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Evaluate the integral.
$\int_{R} \sin \left(x^{2}+y^{2}\right) d A,$ where $R$ is the disk of radius 2 centered at the origin.

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In polar coordinates. The body is in fact represented as a quarter circle of radios. So our radios uh to be equal to a So it's you miss to calculates for the total mass. So uh twitter mass. So we calculated total mass and defined by each of the movements of in asia with their necks. So I'm here. It's going to be called to rule is squared high divided by four. Then I'm moving to phoenicia. I X. Will be called in sick to Dublin cigarette. So it's super pi divided by its suit. You have the cigarettes out to a you are keeps saying squared pizza D. Of the pizza. And this will give us it's the path for group I divided by 16. So you realize that this is Product of two functions. So you apply into patient by parts here. So I why I swill give us hi Sue Danny. How search too A. You have through our cube because squared sita the air the Audi tita. And this is equal to a to the power four Roubaix divided by 16. So the coordinates of radios of derision. So eggs baba. Ex bug radios operation for that to be square roots of hi. Why divided by a which is equal to a divided by soon then why baba? It's going to be square roots of I. X. Divided by a. Which is also a divided by two. So then this implies that the coordinates. So the coordinates the coordinates of the radios. The radios radios Okay, she of direction. Mm hmm. You have a divided by two divided by two as a finer results.

Were given integral of a function over a region and the description of this region were asked to use polar coordinates defying this integral, the integral is double integral over the region de of the function X squared Y d A. Where d is the top half of a disc with a center at the origin and a radius of five. So we can draw our region de using polar coordinates as follows. We have a line that equals pi over two in line fate equals zero, and we knew that maximum value of our is going to be five. We have that. This is the top half of the disc, so we're only looking at data from zero supply antes. It's a disk we're looking at. All radius is less than or equal to five. Great, there equals zero. So our region de looks something like this this blue shaded region which is a semi circle, and we see that and pork ordinance the description of the is set of all ours and potatoes such that our last between zero and five and data lies between zero pie. If we treat our function X squared y as some function of X and Y call it f of X y. We see that X square is a continuous function and why is a continuous function on de and therefore F is continuous on the region de And so since region he is simple It follows that they don't want to grow of x squared y over the region de is well defined and is given by the integral from zero to pi integral from 0 to 5. Uh, polar coordinates X is are who signed data. This is our assigned data squared And why is r sine theta This is or sign data and then for polar coordinates D A is when you are, er data Where the are is that it? In the change of variables this is equal to And since we have that the limits on the integral our constant values We can separate these inter girls using Roubini serum or even his product ID in jiggles. So this is the same as the integral from zero to pi uh, who signed square data sign Peter de Theater times the integral from 0 to 5, uh, are to the fourth er and using substitution for the first integral say u equals co sign of data. Let me have that. Do you is going to be negative. Sign of data de una and therefore and a girl is equal to integral from you have zero is co sign a zero which is one to you of pie. Just co sign a pie which is negative. One of negative you squared Do you times integral from 0 to 5 of art Fourth, you are is equal to two steps at once. So applying the negative we conflict the limits on the integral and then also taking anti derivatives We get 1/3 you cute evaluated from negative oneto one and times one fit part of the fifth diverted from 0 to 5 is equal to 1/3 time's 1/5 is 1/15 times wanted to third is one minus negative one that there is negative one. So this is to times 5/5 is five to the fifth, minus 00 So being five, this is equal to two times five to the fourth over three. Five to the fourth is the same as 25 squared. So 25 times 25 This is 625. So we get 625 times two over three, which is 12 50 divided by three. This is our answer.

So we want to look at the top half of the disc with Senator Origin and Radio Spot two. Here's our disk. That's only the top half in the radius is far. So we know that our radius is going very between zero and five. And we know our fate is in a very from zero to pie. All right, so let's set up this into probably also know that X is equal. Tio Archos on X and two y's equals R Sign of scientific. So are our bounds. You're going from zero to five. Our data bounds. You're going from zero to high that x squared. Why? They also need r r D R D data. So there's going to be r cubed a co sign squared data sign data. All right, so integrating with respects to fate of first let's do data. So we got zero to five zero. Um, sorry, uh, are to the fourth and we're going to get the use of here. Okay, so we're going to say that and I You sequel to co sign data. So what? The sense of being is you squared. Do you Dior? Okay, so they should actually be switched, but it's not that big of a deal. Then we've got, uh let's see here. What was I doing? Yes, this will give us the negative sign. Uh, so you got a negative out here, But when we plug in zero for coastline, we're going to get one and we plug in pi and two co sign will get negative one and then that negative, we'Ll just switch these guys just like that. Okay, So, uh, then we can just rewrite our into girl. Since you is an odd saris and even function go hard to the fourth. We've still got you squared, you tr. So this ends up being, um, to over three zero two five or to the fourth deal. Not too difficult to see how that ends up working out. And then we go to over three. There's one of the five are to the fifth from zero to five. All right, so we got two or three times one over five times five to the fifth. Well, that's two or three times are to the fourth times, five to the fourth. And, um, that is our answer.