I have the following problem: Find the exact length of the curve: $$r = 2(1 + cos(\theta))$$ How should determine the intervals. I used the graph but it is a cardioid and i do not know how to proceed.
2026-03-27 14:02:35.1774620155
Exact length of a polar curve
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Cosine has a period of $2\pi$ and thus your curve does as well.
You can also check this on your grapher if it has a trace feature. You will see that the curve is covered exactly once in the interval $[0,2\pi)$.
You can also calculate some points for various values of theta and see that there is no repetition on that interval.
Therefore, letting $r(\theta)=2(1+\cos\theta)$ the arc length is given by
$$\int_0^{2\pi}\sqrt{\left(\frac{dr}{d\theta}\right)^2+r^2}\,d\theta$$
Can you finish from here?