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Problem 20 Hard Difficulty

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

Answer

$\int_{C} \mathbf{F} \cdot d \mathbf{r}=\frac{17}{15}$

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Video Transcript

So in this question, were given field and were given a position director and were passed to determine the lining. So they feel that we're given. Is this right here? So, experts, why I have plus y minus c j Was he squared k? Another way to right? This is just, um yes. So, experts, why comma y si com Assi square. And the thing is, we want to write so to determine the line. Interpol are basic technique, is we? Her amateurs are curved. Then we write everything in terms of so are curved is given by squared I have plus tq jihad plus t squared que have so are so instead of X. So if we look at our f right here, what we're gonna do is we're gonna four x We're gonna plug in X squared for why we're gonna plug into Cube. And yes, if our X we're gonna plug into squared for driver and a plug in he cubed and foresee. We're gonna plug in Z square. So when we plug these into this right here, But book week what we get is the, uh when we get there's this over here, okay. And then now are empty again. We can write or a pretty in a different way. We can write our of tst squared comment Cube, comity square. And then we lost our prime of tea, which is to t comma three t squared opportunities. So basically, we just took the derivative with respect ity off each terms. This is okay. Again, our technique is the following. We want to find that thought being to grow along. See of that that you are. So this is equal to, uh, now were you right? Everything in terms of tea. So, Ethel, are the dog are prime of TV? Okay, we found out we already know what f a r t is over here. Everything is in terms of tea. We already know what our prime of tea is. Verity determined, that is. And now our limits of integration. Again, we said T goes from 0 to 1. All right, so to find the adult product that just multiply the first term here by the first term here, second term here by the second term here, their term here, right trip. And then what we get? Oh, get this right here. If we distribute here been here. We get duty. Que Plus two d to the power for plus three t to the power five minus three t to the public for plus two. T to the power. And now combining like powers. Well, what we get is So, for example, this is three t to the power five to D to two D to the qualified. We get five feet of the power five here. We have to teach the powerful or minus three t to the power for it's minus one t to the power for and to t Cube. All right, Now all of these airpower staying the integral is pretty easy. So this is just a regular integral. And the limit points is from the upper limit is one of lower limit zero. Now we plug these in, we get 1/2 plus 5/6 minus 1/5. This is when we plug in one for teeth. When plug ins era 40 we get zero plus zero minus zero. All right, so we have 1/2 plus five six minus 1/5. And what we can do now is this Think of a common factor. Are common denominator a common denominator by but least common multiple that comes to my mind is 30 So we're gonna multiply 1/2 bite you give are resting We're gonna multiply 56 by five Divided by five that we're gonna multiply 1/5 by six, divided by six So then what? We get best 15 plus 25 burns 36 And that's 40 minus six, which is 34 divided by 30 and then 34 30 share of common factor of two. So are lining to girls just 17 divided by