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I am in Calculus II and am learning of the integration of polar functions. I understand the basic concept, that you are taking sectors of a circle from the pole to the function. However, I don't understand how that works for a function that does not contain the pole. If you were to try to take the area of such function (for example, r=cosθ), wouldn't the sectors of the pole take the area outside the circle and not inside the circle? Thanks

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No because a line drawn from the origin at an angle $-\frac{\pi}2\le\theta\le\frac{\pi}2$ from the initial line would always be in contact with the inside of the circle mentioned. This means that any sector created would be formed within the drawn circle and hence the area found is within the circle.

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