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What goes wrong with this argument?

Suppose I have a curve $\gamma: [a, b] \rightarrow R^n$. We express this in polar coordinates as $\gamma(t) = (r(t), \theta(t))$. The derivative of $\gamma$ is just taken componentwise, so $\gamma'(t) = (r'(t), \theta'(t))$. Now the length of $\gamma$ is given by $\int_a^b \| \gamma'(t) \| \; dt = \int \|(r'(t), \theta'(t))\| \; dt$. In polar coordinates, the norm of $(r'(t), \theta'(t))$ is simply $r'(t)$. It follows that the length is just $r(b) - r(a)$, but this is clearly wrong. Where are the mistakes?

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