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Evaluate the line integral. JcF-dr, where F(x,y)=xy i+x j 15 and C is given by Pts r(t) = sin(t) i+ (1+t) j, 0 <t <t...

Question

Evaluate the line integral. JcF-dr, where F(x,y)=xy i+x j 15 and C is given by Pts r(t) = sin(t) i+ (1+t) j, 0 <t <t

Evaluate the line integral. JcF-dr, where F(x,y)=xy i+x j 15 and C is given by Pts r(t) = sin(t) i+ (1+t) j, 0 <t <t



Answers

Evaluate the line integral $\int_{C} \mathbf{F} \cdot d \mathbf{r},$ where $C$ is given by the vector function $\mathbf{r}(t) .$
$$\mathbf{F}(x, y)=x y \mathbf{i}+3 y^{2} \mathbf{j}$$
$$\mathbf{r}(t)=11 t^{4} \mathbf{i}+t^{3} \mathbf{j}, \quad 0 \leqslant t \leqslant 1$$

So the value is a line to crow. And, uh, here way, computer. Looking like the previous problem, we come here for far off T That would complete our prime teeth. And we take the we take the dot product integrated and Ortiz a lower point of seo in hwa. So access teach you. So we have thiss Why's that fifty square? Because I see even functions or negativity school cause and in fifties were no cause and he scores the something exorcised force. The road is three t square minus two tea one so integral from zero to one than their products or street he square scientific You minus two t co site is square plus t to the fourth Stickley and then tied the rift. You should be able to see that is straightforward You Some problems take you because you that despises you. So this this should be Detective Cho ce I kill and summer. This should be that if sai t square pastie to the fifth or five. We make sure this derivative to teach those from an active and this one's pretty square. Yeah, So you probably one you get minus co sign one minus sign one plus one over five. You're probably zero you can necked if one my zero plus zero. So this one's reprocess Want aliens or six over five minus coast? I want minus sign one.

Big You wanna go exit the years on See, Is the line segmental to come over? And farmers you know to to Hamas in Obama too. So that is fine. X X is equal due to plus through minus doing toe TV it is it will do to you Bicycle one plus zero minus one into musical one minus t their physical zero plus tu minus zero in tow is going toe duty. So my DS is giving us It's Dashti was well plus by Dashti Foursquare plus the Dashti who was were DT. So notice find what is X stashed e? It is zero plus minus one whole square. Pleasant Dashti is two holes were so I get the answer Ruto by de Deep Integral Alex that the yes can be return is what is x two? That is duty into Ludo five dd It is equal to a full file. DDT is equal to four ruled by these goodbye to And what is the integration say zero toe to from taking those add value 0212 zero Due to So is it is a weirdo to root five, even 24 It is a good way it rude fight

Okay, folks. So in this video, we have this line into grow that we need to evaluate. We have this integral along see of X minus y plus Z minus two. And where we're doing the integral with respect to s the variable s. And, um so the way we're gonna do this is we're gonna as usual when you convert this DS into something with respect to T. If you remember, the US is really just the x d t squared plus d y d t squared plus d c g t squared multiplied by d t outside of the square root. And so now we're gonna evaluate X and Y and Z evaluate the derivative with respect to time, So excessive function of tea is just tea. So so ex prime of tea is one and ex prime. I mean, why prime the tea is minus one, because why is one minus? T and Z is a constant. So when you take the derivative of the of a constant with respect to a variable, we get zero z prime of t zero. So we're gonna cross this out because it's zero. Now we put this in here. We get the s equals. Um, square root of one squared, which is one and a negative. One squared is also one multiplied by DT. So we have route to DT. So we're gonna plug this back into the integral, which is right here, and so are integral becomes X as a function of time is t minus. Why? Why is one minus t plus Zied C is a constant C is one and then minus two. So this is the Inter Grand. The DS becomes now routes to DT. All right, that's good. So we're gonna evaluate what this whole thing is. So it's, um t minus one plus T plus one minus two route to DT. Okay, so now I'm going to do, um What's this? Minus one plus one. Okay, so we have to t minus two. So two of T minus one, right to t minus two easters, two of T minus one times route to. So we have to root two t minus one. DT. Okay, so this is a constant. We're gonna pull the constant out. So we have to root two of of the inter grow of T minus one D t, which is gonna be to root two of what this t squared over two minus t. Um, evaluated are the two limits. What's the first limit? Well, the first limit is when x y Z is 011 And the second limit is when x y Z is 101 Well, when x at the lower limit, X zero when x zero, t zero. Okay, so we have t equals zero. What about the second? The upper limit? Well, the upper limit says that. Why is zero right? What? When? Why is zero that means one minus t. Because why equals one minus t zero? That means tea is one. Okay, so let's plug in the two limits of integration. We have to root two of 1/2 minus one. That's the 1st 1 minus zero minus zero. Okay, so we have this minus 1/2. This is gone because zero, um, this, too. And this to cancel out minus route to is where we have left. All right. I think that's it for this video. Thank you very much. Let me raise this arrow for you. I wanted to draw a narrow, but I failed. Okay, so we have negative route to That's it for this video. Thank you.

So you value it is in the grove. Well, we first paramour tries a curve and this one is easy access t wise to tea these pretty and he goes from zero to one. So this way you permit tries you get access key. Why's Tootie Easy Street is or sixty square and ps his square root off one plus one square past you square plus three square So the square root of fourteen TT So it's a square with a fourteen tp and we use a U substitution. So sixty square horse you twelve he host you So he goes from zero to one. You should goes from zero to six. He took the hue square, the fourteen titties over turf to you So that should give us this number. Integrate It will use and hide the relativists theory to you and the provinces six and zero. So should I eat six minus zero, Which is what? And that should be the answer


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