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Problem 37 Hard Difficulty

Given the following vector fields and oriented curves $C$, evaluate $\int_{C} \mathbf{F} \cdot \mathbf{T} d s$
$\mathbf{F}=\frac{\langle x, y\rangle}{\left(x^{2}+y^{2}\right)^{3 / 2}}$ on the curve $\mathbf{r}(t)=\left\langle t^{2}, 3 t^{2}\right\rangle,$ for $1 \leq t \leq 2$

Answer

$$\frac{3}{4 \sqrt{10}}$$

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Video Transcript

All right. Over 37. Given the following vector field, Find the line in a girl for our of t equals T square comma three t square immediately. Let's just calculate our priority. Okay? Line And the girls you're from or to of EFTA uh, our primary t do you t And for our X and why we're gonna plug in t Square three t squared. And we have our tea. So everything we need to solve the sign in a girl plugging in everything we get integral from one of two of t squared three t square over tend to three or two. There are six dot That was to t comma 60 and a girl from one to over. Turn Kurt of 10 t 26 to Cuba's 16 que Now this simplifies to in a girl from one to, uh, t negative three d t We evaluate this or get too scared of turn t to negative two over negative too. If I waited from wanted to playing in our numbers, it's cancel. Get 1/4 minus one, which is a good a 3/4 square to 10

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