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Find the length for the following curves.

$$

\mathbf{r}(t)=\langle 3(t), 4 \cos (t), 4 \sin (t)\rangle \text { for } 1 \leq t \leq 4

$$

$\int_{a}^{b}\left\|\mathbf{r}^{\prime}(t)\right\| d t=15$

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Okay, so we have to find our cling. So that's our G is equal to three G for coastline empty and for sanity, the 60 derivative of our to get R T is equal to Well, it's this nine. Oh, no, I think it's this three before sign of tea and Oracle sanity. Now let's take the magnitude of Artie. Don't get this. Isn't the square root of nine plus 16 sine squared? That's cool science. We'd cancers. We could 16 plus nine. That's 25. So that's the secret. If I Okay, now, let's pick the integral from 1 to 4 or five that used me just five c evaluated a foreign ones, which gives me Christine.

University of California, Berkeley