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A lima莽on is a type of polar curve that is defined by equations of the form \(r = a + b \cos \theta\) or \(r = a + b \sin \theta\). This curve is named after the French word for snail, as its shape often resembles a snail shell.
Depending on the relationship between the constants \(a\) and \(b\), a lima莽on can have different forms:

• If \(a < b\), the lima莽on has an inner loop.
• If \(a = b\), the lima莽on passes through the pole once, becoming a cardioid.
• If \(a > b\), the lima莽on is dimpled without a loop.

In our specific example, \(a = 2\) and \(b = 4\), since \(a < b\), the lima莽on will have an inner loop. Understanding this helps determine which areas of the graph need to be considered when finding areas, especially in relation to the pole (the origin in polar coordinates).
This forms the basis for solving the exercise related to the area outside the inner loop of the given lima莽on equation.

Calculating the area enclosed by a polar curve involves integrating the function that defines the curve. For a curve given by \(r = f(\theta)\), the formula for the area \(A\) is:\[ A = \frac{1}{2} \int_{\theta_1}^{\theta_2} r^2 \, d\theta \]This formula essentially calculates the sum of infinitesimally small triangular wedges that make up the area under the curve.
For the example of the lima莽on \(r = 2 + 4 \cos \theta\), the problem requires finding the area outside the inner loop. This involves:

• Determining the points where the inner loop intersects with the pole (i.e., when \(r = 0\)).
• Setting appropriate limits of integration to cover only the region outside the inner loop.
• Finally, setting up and computing the integral \(\frac{1}{2} \left( \int_{0}^{\frac{2\pi}{3}} (2 + 4 \cos \theta)^2 \, d\theta + \int_{\frac{4\pi}{3}}^{2\pi} (2 + 4 \cos \theta)^2 \, d\theta \right)\).

These steps ensure the accurate computation of the area for the desired section of the polar curve.

To simplify the integration process in calculating areas of polar curves like the lima莽on, trigonometric identities are crucial. For the exercise, simplifying \((2 + 4 \cos \theta)^2\) utilized such identities. Let鈥檚 see how:- The expression \((2 + 4 \cos \theta)^2\) expands to \(4 + 16 \cos \theta + 16 \cos^2 \theta\).- The identity \(\cos^2 \theta = \frac{1 + \cos 2\theta}{2}\) helps in further simplifying expressions involving \(\cos^2 \theta\).Using these identities simplifies the integrals by converting trigonometric squares into simpler expressions, which are easier to integrate. Understanding trigonometric identities is essential because they provide a method to transform complex trigonometric functions into combinations of simpler functions. This transformation is particularly useful when integrating over specific intervals as part of solving polar coordinate problems.