Integrate $f$ over the given curve.

$f(x, y)=x^{2}-y, \quad C: \quad x^{2}+y^{2}=4$ in the first quadrant from (0,2) to $(\sqrt{2}, \sqrt{2})$

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Harvey Mudd College

Okay, folks. So in this video, we're gonna take a look at problem number 30. Um, the way we're gonna do this is by by recognizing that we have a circle during the great over. So whenever you have a circle, you can you can always think of, you know, parameter rising X And why, um, with a new variable t or theta. But I'm gonna write t So we have accepted was to co sign of tea. And why of t equals. You know, I'm just defining you function here, um, to make our lives simpler when we're evaluating the integral. Okay, So why equals two signing t The reason this is two and not one or whatever is because we have a circle with a radius of two. Okay, so So let's figure out what Ds is, so we can pluck that in later. We have DS equals, um, ex prime affects. Excuse me? No ex prime of X. We have ah, ex prime of t squared. So and plus why Prime of t squared multiplied by d. T. I hope you all know that. And, um if you value eight ex prime, you get negative to sign t I. And if you evaluate why a problem, you get too close. I and you square those two things and you add them up, you're gonna get before multiplied by DT. So that's just two DT. And so now we're going to Ah, Now we're ready to plug in everything into our integral, Which is that the integral long sea of F. D. S. And after you plug in everything you get four. Who's i n squared of D minus two. Sign of tea multiplied by two D t from zero to pi over four. The reason this number where I got this number from is, um if you ah, if you look at the problem, you're integrating from 0 to 2. Root to root two. And this is where route to route to is is 45 degrees. Okay, so 45 degrees is really just pi over four. So that's that. That really you know, the way you get this piper for it's going to take a little bit of, I guess, practice. Um, if you're really familiar with trigonometry, then you can just look at this route to route to and just immediately realize that this point is simply the point where the angle is pi over four. Anyway, I am going to split. Um, this inter crawl into two parts. So we have We're going tohave eight co science garden T d t minus four. Signed T D t. Okay, this is from zero to pi over four. Just as before. The same thing with this zero to pi over four. And this is going to be now if you, uh if you remember, you're too. Not much identity Coastline square. It could be written as one plus co sign to t over to. And if you, uh, plug that into this this expression this integral right here, you're going to end up with four t plus sign of two t over to between zero and pi over four. That's the first integral we have. For the second interval, we have plus four co sign t between zero and power over four. And if you crank out this algebra, you're gonna get pie place to plus two route to minus four. And when you add them up, you get pi plus two times fruit to minus one. And that's the answer for this problem from number 30 and we're done for this video. Thank you.

University of California, Berkeley