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Use Green's Theorem to evaluate, sketch region R: tan' (x) dx Jxdy;C is a rectangle with vertices (1.0),(0,1).(2,3),( 3,2)...

Question

Use Green's Theorem to evaluate, sketch region R: tan' (x) dx Jxdy;C is a rectangle with vertices (1.0),(0,1).(2,3),( 3,2)

Use Green's Theorem to evaluate, sketch region R: tan' (x) dx Jxdy;C is a rectangle with vertices (1.0),(0,1).(2,3),( 3,2)



Answers

Use Green's Theorem to evaluate the line integral along
the given positively oriented curve.
$$
\int_{C} y e^{x} d x+2 e^{x} d y
$$
$C$ is the rectangle with vertices $(0,0),(3,0),(3,4),$
and $(0,4)$

So use all Christians in the valley with this light grow. So by current serum, this will be the seventh thie area. And to grow off the ripped half of this with respect to X, so to eat with X minus the river till this was respectful. Why? Which is eaten with X, uh, are about all right, ta here. So either the ex d a. So this will be isa rocked. M go from zero zero to three zero three four to the oh four. So is simply X from zero to three. Wife from zero to four on the beach with X the the X from zero to three. Wife zero two four. So if it was ex from zero to three So we have, uh, these three minus one d Why And this is a constant always simply have the muted prized by four

I have a problem for 42. So we got Ah, simplify this using green serum. Okay, So this are integral. Combining the two integral roles substituting our limits simplifies to you. Go from 0 to 4. Why squared minus six? Do you? Why? And four Why? Cubed over three minus six Y evaluated from 0 to 2 because this negative 4/3.

Okay, So in this video screams from the vine to call this right here. So we're giving Cosan wanke X plus X squared sine Why? Why were also determined? Lining true. And we're given that c is falling rectangles. So we go from 0050 from 50 to 5 to from 5 to 202 purposes excellence. Zero boys. All right, so the first thing we have to do is you have to take so we're gonna label on P's and Q's and then we're gonna take the derivative silicon in t derivative of P break here. Sorry. The derivative off Q. Which is right here with respect to X and the derivative off peak right here with respect to y So we're gonna get for its or the derivative Wouldn't respected exit exclaimed, Same wise just to let's fine wife and there. But it would respect Why, of course, in wise, just negative sign why and never minus on the outside that becomes plus sign. No, you're free. Look right over here we see that so we need our limits of integration. So we noticed that why goes from do you wrote it too, while X goes from spirit of flight. So our limits of integration for why is from 0 to 2 pi r limits. So for why are in the middle of integration from 0 to 2 by my minute into creation for the exits from Okay, so now we only have to compute a double integral. So the integral of two X sign Why? With respective eyes. So two X will be constant because there's no why in that, But the integral of sine life is negative course. And now that you represent out business, got that through a bit of a break here. Now, for the derivative off plus sign, right, That's just negative coast on our limits of integration or from 0 to 2. So we're gonna plug in to for why and then syrup forget to for a while this right here. Minus that, we're gonna plug in zero for wine. But we know that co science does one 1st 11 For now we get this and we'll ride over here and now we simply take the derivative respect effects. So the derivative with respect axes just X squared, divided by two. So this is why this two will cancel this to write year cancels. We're left with negative X squared co sign too. Plus now the integral coastline to there's no extenders but we know that the integral So going to treat it like a constant So we just caught a ranks integral to X is just X squared, divided by two. So the two cancel or just left the next square integral born is just X Okay, so now we're gonna plug in 54 x rather than against us right here They were plugging zero for six and then all the terms will cancel. So we have negative 25 co sign to minus five course items. Just negative. Vertical site, too. Then we have 25 plus five, which is 30. But we can do is we can pull out common factor birdie better rearrange. So we got one minus co sign too. So our final answer is 30 times one minus co sign, too. So this is the value off the light into cold determined Green's field

So compute this this integral character and using currency here as through a picture first. Zero zero. You're five zero. You're five four two zero four. So it kind of goes this way. And, uh Okay, so zero zero two five zero over. So that's give those curve unnamed. C one C to cease received for and we have to Paramount tries off them. So see, one will be ex ecos five t wife was here. She from zero to one seat You will be exit was five. I was forty. See from there at once. See story again we use a standard line dramatization will be ex course Caesar's from here to here. So five miners, five t Why equals four? She goes from zero to one and C four will be X cause zero. Why equals four minus forty She goes from zero to one. So let's do it through the first part Computer directory. What is into grow? See one. Why scored the X plus X square? Y t y That's his fight y zero. It's why zero? We don't have to come with the white part extra because y zero uh, this is zero. This is also zero. So this integral just be zero. So this part, it's easy to compute as computer see two and this has exit. Who's five? So the excess Ciro So this term you we don't need to compute better use a different color and x square. Whitey was try to figure out this Chico's from zero to one X Square is twenty five times why wise forty. So is one hundred eighteen and the wise forty. So it's four hundred petey and every winter go we got two hundred he square planning What? So we have two hundred years and we never see saree y squared the explosive explorer wife Why in Sicily we have X equals five way have wife who's a constant so d y zero We don't have to compute this part and the y squared DX why Square is sixteen So he goes from zero to one y scores sixteen and the access our next five tt So we have next eighty and Integral found zero tow what's left of eighty intimacy for because x zero this y squared because access zero this party zero And this the axis also zero So we're interpreting nothing and that means getting zero. Also final answer. So we add all four together we get into crow. See y squared T X plus x square Whitey, Why should be Oh, four number added together should be one twenty. So that's what we get By using the direct master Now, I want you home by using the greatest era So using a green serum, we'LL have x y squared So we're y squared the X plus x square Why the wine? So it becomes integral of the region in close by sea and the derivative off this with respect. Why? Which is X square minus a derivative off this component with respect to X, which is zero. I'm sorry. That's wrong. The river tees off this part with respect to X so which should be to ex wife minus derivative of this part with respect to why you should be too wide. So let me let me write out ofthe general former PT X plus cute, because Marshall Marshall X minus people. So this should be this one with respect to X minus. This one was respectable wife. Now what is a region is a rectangle from from zero to five extra goto fire wife on the order form. So the extra y that's from zero to five wife from zero to four. So we're taking the anti taru to with respect to X first. So x square Why minus two ex wives from zero to five. So it probably x it was five. Twenty five y minus ten. Why? And that would probably actually four zero zero. We have zero four, fifteen wives. Why? And we've got fifteen twice square over too. The entire the route from the other for the problems. You don't got nothing. So we just have to probably for now, fifteen times sixteen over too, which is one twenty. And we get the same answer as we get by computing computer directly. There also is this also example off the grazie.


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