Area in Polar Coordinates

Find the areas of the regions in the polar coordinate plane described in Exercises 47–50.

Inside the cardioid r = 2(1 + sin θ) and outside the circle r = 2 sin θ

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Identify the curves given: the cardioid is described by \(r = 2(1 + \sin \theta)\) and the circle by \(r = 2 \sin \theta\).

Determine the region of interest: the area inside the cardioid but outside the circle means we want the points where \(r\) lies between the circle and the cardioid, i.e., \(2 \sin \theta \leq r \leq 2(1 + \sin \theta)\).

Find the points of intersection by setting the two polar equations equal: \(2(1 + \sin \theta) = 2 \sin \theta\). Simplify this to find the values of \(\theta\) where the curves intersect.

Set up the integral for the area between the two curves using the formula for area in polar coordinates: \(A = \frac{1}{2} \int_{\alpha}^{\beta} \left( r_{outer}^2 - r_{inner}^2 \right) d\theta\), where \(r_{outer} = 2(1 + \sin \theta)\) and \(r_{inner} = 2 \sin \theta\).

Determine the limits of integration \(\alpha\) and \(\beta\) from the intersection points found earlier, then write the integral explicitly as \(A = \frac{1}{2} \int_{\alpha}^{\beta} \left[ (2(1 + \sin \theta))^2 - (2 \sin \theta)^2 \right] d\theta\).

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주요 개념

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Polar Coordinates and Graphing

Polar coordinates represent points using a radius and an angle (r, θ) instead of Cartesian (x, y). Understanding how to graph curves like cardioids and circles in polar form is essential to visualize the regions described by equations such as r = 2(1 + sin θ) and r = 2 sin θ.

Area Calculation in Polar Coordinates

The area enclosed by a polar curve r(θ) from θ = a to θ = b is given by the integral (1/2) ∫[a to b] (r(θ))^2 dθ. This formula is fundamental for finding areas bounded by one or more polar curves by setting appropriate limits and subtracting overlapping regions.

Determining Intersection Points and Integration Limits

To find the area between two polar curves, it is crucial to identify their points of intersection by solving r1(θ) = r2(θ). These intersection angles define the limits of integration, ensuring the correct portion of the region is calculated when subtracting the inner area from the outer.

Area in Polar CoordinatesFind the areas of the regions in the polar coordinate plane described in Exercises 47–50.Inside the cardioid r = 2(1 + sin θ) and outside the circle r = 2 sin θ