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Problem 8 Medium Difficulty

Evaluate the line integral, where $ C $ is the given curve.

$ \displaystyle \int_C x^2 \, dx + y^2 \, dy $, $ C $ consists of the arc of the circle $ x^2 + y^2 = 4 $ from $ (2, 0) $ to $ (0, 2) $ followed by the line segment from $ (0, 2) $ to $ (4, 3) $


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Jeremy S.

November 14, 2019

How did you know the bounds for t in C2?

Video Transcript

So here a star by as This Star by Paramount tries those Cruyff and then combine together. Let's call this our part of the Circle Sea one on the sly segment C too, and has tried to Paramount Rice both of them. It goes from to zero, which is on the except that it's two zero two along the Y axis. So this is like the part of Circle First Quadrant. So you use a standard per mature ization access to co sign t wise to society. And he goes from zero two pi over, too. What is seat you Setu? We can use a standard per mature ization off the line and, uh, X should be forty wise to plus tea and sure, plus t and T goes from zero to one. We just do the integral arms separately and added Together, let's open another page So into grow they say that's to see one of our first. It goes from that This this is a parley with deal with C one first she goes from zero to pie over too Ex squares for coz I scurti the ex is negative to sigh Inti dt so that to sight see plus y square is for science security. The wise is it wise to co sign. So let's simplify this purse. Zero two pi over too, because there are eight that we have sign the sai skirt he co signed minus side d ho sai squared. So that's tried, Tio, solve this integral. See if there's any easy way or we just have to do it separately. Well, we showed them we can do a u substitution. I'm just feeling hours away. Wish my make this be easier. I guess we can just do it separately. There are no pie over to science square T co sign minus zero Tau pi over to side B coz I scurti for the press probably without you it was scientific. And for the second part, we let you horse Costa and this hot tea you'LL be negative sign titty And for this part to you co sign t So let's see aliens continue No eight I forget this It always should have eight times into grow zero two pi over too. Ah Sai squared! So we have awesome, sir. For the first part we do the substitution. So science chorus you square you square and, uh, cause and it is you and fun. Twenty goes from zero to pirate You signed. He goes from zero to what? And there's a second part coz I scored Tio issues Claire scientists next website It it is to you. And when he goes from zero to high over two coastline Chico's from one to zero. Yeah. So this is who can sew and that we should get a zero for this part on DH. Yeah, I'm not sure if it's actually easier way to do. It seems I do a lot of work to do to just to figure out this part and the seven price of pulling over into grow. So that's trying to do so. So the integration will just be on the second part on, uh, let's just do this for the seat you part xs forty What is two plus two and the Inter crows Franz From zero to one. Ex squares or sixteen t square The access to the sea Plus why squares too plussed he's square and the wide sea it is just It's just what? So it's just here. So zero to one, we have thirty two away with him is something at the city. Know what I'm doing here? This is not right. X is forty. So x square sixteen t score And the axis Just our forty. So this was sixty four T square, plus Andi Mitchell. Delighted. Delighted His one. So the wise teacher. Why square? Okay, so this square plus forty plus form. So we combined together sixty five square plus plus forty plus four. And so first time we got Cube over three, and we problem once. We got one of those street city five over three, plus your t square over too. So we find one you want to. So I want two times for us, too. And into grow four from zero to one. Keeps us for yes, for prostitutes. Six. Eighteen over three. So it's should be eighty three. And you can verify the answer yourself. His looks at her very lengthy computation, and I might make some arrow here. Hopefully the concept is clear