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[10 pts:] Evaluate the line integral (sin 2 + y? )dz + (x + e" )dy; where C is the positively oriented boundary curve of the region D : 1 < 22 +y? < 4...

Question

[10 pts:] Evaluate the line integral (sin 2 + y? )dz + (x + e" )dy; where C is the positively oriented boundary curve of the region D : 1 < 22 +y? < 4

[10 pts:] Evaluate the line integral (sin 2 + y? )dz + (x + e" )dy; where C is the positively oriented boundary curve of the region D : 1 < 22 +y? < 4



Answers

Evaluate line integral $\oint_{C}(y-\sin (y) \cos (y)) d x+2 x \sin ^{2}(y) d y, $ where $C$ is oriented in a counterclockwise path around the region bounded by $x=-1, x=2, y=4-x^{2},$ and $y=x-2.$

The value is a line to grow. First we paramour tries the lie excessive force one plus for my wife's reason. Three. T Why? Who's easy? Host Tootie? He goes from zero to one So this inter crow should be here on zero two one c square the exes Streett plus X Square The wise TT Plus Why square This's TT and we just call out the term together way had twelve t square plus ninety square plus sixty plus one plus two square. So what do we wear? Twenty three t squared plus ah, sixty plus one. So we should have twenty three times one over three kilos. His teacher over three we probably want plus six times one over ritual, which is three plus one. So this is for For which is twelve over three so thirty five over three

Evaluated this lying to grow So So we have to part with all this C one called Setu and the repairable tries both C one should be actually host t y ho zeros he caused t and he from their toe. One c two ex course What minus t Why cause t The horse won Krusty under t goes from zero to one and uh let's see So it's open another pages too to compute that the C one part we have integral from zero to one So what is y processed? The axe is ex pluses to tea Hi pretty y zero x plus y eyes t on easy stt So this party's should be easy so t square and we caught him once So we should have won come here to see two apart Why posses one plus two tea The access negative dt explosive is too the wise x plus wise one and he's he's teaching that keeps us zero to one. Ah, sory. Minus voice to minus T t ondas Shoot keeps us to minus two times one over two. It was T square over to plough in one two times. One over to this one I know we feel Wait and, uh, give us one. So the original in the grocery by's some of these two, which is one process one is two.

Iterated into the So let's go ahead and start this one from integral from zero to pie over two of injury will of zero tax of X sign. Why do I d x? So we first integrate with her specs toe. Why? Because that's the first differential that comes into play here, So X is a constant. So the integral of sine of why is negative x co sign of why? And we integrate this from zero to X. You gotta dx so fundamental theorem of calculus says that this will evaluate too negative x co sign X plus, um ex co sign Europe. Okay, All right, so this is really just ax minus X coz I knw axe. Right, So now we have to integrate this and this first into girl is not too difficult. Alright, Khun right this into girl and actually spoke. It's what the integration. So we should, uh, should do here. So this first into girls is an integral from calculus one, and the second one most likely is an integral from calculus to depending on where you took it the second one, we're gonna have to use integration by parts. So let's write our integration by parts here. So we let you equal X Let Devi equal co sign axe DX. We let you do you know, we differentiate you, we get DX and the anti derivative. The co sign is sign. All right, so let's write this into girl out. So this is X squared over two from zero to pie. Over. Too minus. This is going to be the integration by parts stuff X sign ex evaluated from zero to pie over to minus integral of zero to high over too of sign of ax on. There's a DX in here. The erase this stuff to make it look nice for you guys. Sign of X D X. Okay, so evaluating this guy here, we're going to get hi squared over eight minus. When we plug in zero, we'LL just get a zero over here. We're going to get high over two. Sign of high over too. Minus zero times Signe of zero and then minus Well, what is the antidote of sign? It's negative. Co signs of this becomes positive co sign of axe evaluated from zero to pie. Over too. So some cancellation here. These are both going to be zero. I'm sorry. This is not going to be sure. This is just going to be one sign of pie. Over two is one. So So I end up being pi squared. Ovary minus pi over too. Minus co sign. Hi Over too. Um minus co sign zero. Right. So co signed a fire to zero. No sign of zero is just one. So our final answer here is going to be higher squared over eight minus pi over. Too less one, and that's it.

So you see Quincy, evaluate this so he will be. He will host the area into growth. The review of this term with respect to X which is too minus the ripped you off this time with respect or why which is one. So this will be area off given region. So we just have to Computer area itself is between Why whose x square and X equals y square. So that's why those x square and this X equals y square, We know intercept zero zero and one one So into Chris one, uh, say why form the lower one is x squared the opera one square it off x and x from zero to one d y t x So here are a square root of X minus X security x So absolute one off to irritate and hydro tears are actually three over too divided by three over to arm ut Private. You're history minus X cubed over three and probably zero years, you know, So we just have the problem of one. It gives us to over three minus one over three and therefore won over three


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