44–49. Areas of regions Find the area of the following regions.
The region inside the cardioid r=1+cosθ and outside the cardioid r=1−cosθ

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First, understand the problem: we need to find the area of the region that lies inside the cardioid given by \(r = 1 + \cos\theta\) and outside the cardioid given by \(r = 1 - \cos\theta\).

Determine the points of intersection of the two cardioids by setting their equations equal: \(1 + \cos\theta = 1 - \cos\theta\). Solve for \(\theta\) to find the limits of integration.

Recall that the area enclosed by a polar curve \(r(\theta)\) from \(\theta = a\) to \(\theta = b\) is given by the formula \(\frac{1}{2} \int_a^b r(\theta)^2 \, d\theta\).

Set up the integral for the area of the region inside \(r = 1 + \cos\theta\) but outside \(r = 1 - \cos\theta\). This will be the difference of the areas enclosed by the two curves between the intersection points: \(\frac{1}{2} \int_a^b \left[(1 + \cos\theta)^2 - (1 - \cos\theta)^2\right] d\theta\).

Evaluate the integral over the interval found in step 2 to find the area of the region between the two cardioids.

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Polar Coordinates and Graphing

Polar coordinates represent points using a radius and angle (r, θ), which is essential for understanding curves like cardioids. Graphing these curves helps visualize the regions bounded by the given equations, making it easier to set up integrals for area calculation.

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Intro to Polar Coordinates

Area Between Curves in Polar Coordinates

The area between two polar curves is found by integrating the difference of their squared radii over the relevant angle interval. Specifically, the formula involves integrating ½(r_outer² - r_inner²) dθ, which accounts for the region inside one curve but outside the other.

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Intro to Polar Coordinates

Determining Limits of Integration

Identifying the correct angle bounds where the two cardioids intersect is crucial for setting up the integral. These limits ensure the integration covers only the region inside one cardioid and outside the other, avoiding overcounting or missing parts of the area.

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