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Use a calculator or CAS to evaluate the line integral correct to four decimal places.

$$\int_{c} \mathbf{F} \cdot d \mathbf{r},$ where $\mathbf{F}(x, y)=x y \mathbf{i}+\sin y \mathbf{j}$$ and $\mathbf{r}(t)=e^{t} \mathbf{i}+e^{-t^{2}} \mathbf{j}, \quad 1 \leqslant t \leqslant 2$

$\int_{C} \mathbf{F} \cdot d \mathbf{r}=1.96334$

Vector Calculus

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So in this question, we're asked to use a program or calculator in order to determine the line integral. But before that, we have to set up our into grow that rocketed to a calculator or a program t determine the value of the range. So we're given that, uh I mean, it's just x y I have assigned jihad, and we're also told that the position vector is e t. I had Plus is the power of negative t swear jihad where t varies from 1 to 2 now, first of all, we want to write everything in terms of so for x. So in f of x and Y or expert and a plug in you did the ballot 54. Why? We're gonna plug in either the power. Okay, So extension allies just either the tee times eat of the power of Negative t Square, and then you we just at the experience. So what we're gonna get is be to this under Skinner, so we quickly do it. You just get me to the power T minus square, then sign of why. So instead of wired a plug in even even negative square to become signed all right. And then our pride we just take the derivative of are so you know, the tea becomes remains Even the tea we need to the negative square becomes negative. Two teeth e to the negative T square. So we should we should know how to do these into girls These derivatives from are so now we determine this too. And then now we proceed to determine are lining to grow. So another way to write the line integral is like this where everything is in terms of tea and so the integral from wanted to So we're going to take the dot product of asked at our price. So that's basically done by multiplying the first term here by the first term here, second term. So what we get gazing row foam Wanted to eat to the T minus T square turns you to the T plus sign into the negative t square times negative to TV to the negative to square. Thank you. So we're gonna So what we can't is the falling. So we just add c two TV dynasty squares. We'll get to t ways to swear And, uh, there is no other simplification. We can do so again. The problem tells us that we can use software, so we use software we get. The final answer is 1.96334 The whole idea is to get to this point, and then we're not gonna race their time determining this integral, because the question says, just poured it into a calculator or some algebra software get divine.