# How do you find the area between the loop of #r=1+2costheta#?

Please see the explanation.

This reference Area with Polar Coordinates does a very similar exercise.

The equation from the reference is:

The integral for the area of the loop is

I am going to let you do the evaluation.

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To find the area between the loop of the polar curve (r = 1 + 2\cos(\theta)), you need to calculate the definite integral of (r) squared with respect to (\theta) over the appropriate interval. Since the loop repeats every (\pi) radians, you would integrate from (\theta = 0) to (\theta = \pi). The formula for finding the area between a polar curve and the origin is:

[A = \frac{1}{2} \int_{\alpha}^{\beta} [r(\theta)]^2 d\theta]

Where (\alpha) and (\beta) are the initial and final angles, respectively, and (r(\theta)) is the polar function.

So, for (r = 1 + 2\cos(\theta)), the area would be:

[A = \frac{1}{2} \int_{0}^{\pi} (1 + 2\cos(\theta))^2 d\theta]

You would then evaluate this integral to find the area between the loop of the curve.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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