Question

Find the area of the regions bounded by the following curves. The limaçon \(r=2-4 \sin \theta\)

Short answer

Answer: The area of the region bounded by the limaçon is \(9\pi\).

Step-by-step solution

Identify the range of angles required for the curve

The first step is to analyze the given equation \(r = 2 - 4\sin\theta\). We notice that when \(\sin\theta = 0\), our \(r = 2\). This happens when \(\theta = 0\) and \(\theta = \pi\). As \(\theta\) increases from \(0\) to \(\pi\), \(\sin\theta\) increases from \(0\) to \(1\) to \(0\). Since the term "\(-4\sin\theta\)" varies between \(0\) and \(-4\), we only need to find the bounds of \(r\) for \(0 \leq \theta \leq \pi\).

Determine the bounds for the integral

Based on step 1, the required range of angles is between 0 and \(\pi\). To find the limits of integration, we look at the value of \(r\) at each point. At \(\theta = 0\) and \(\theta = \pi\), we have \(r = 2\) and \(r = -2\) respectively. Therefore, the limits of integration for the angle range are 0 and \(\pi\).

Set up and evaluate the integral

We now set up the integral to determine the area of the bounded region within the limaçon. The integral is given by: $$A = \frac{1}{2}\int_0^{\pi} (2-4\sin \theta)^2 d\theta$$ To evaluate this integral, first expand the square: $$A = \frac{1}{2}\int_0^\pi (4 - 16\sin\theta + 16\sin^2\theta) d\theta$$ Now evaluate each term separately: $$A = \frac{1}{2}\left[\int_0^{\pi} 4d\theta - \int_0^{\pi} 16\sin\theta d\theta + \int_0^{\pi} 16\sin^2\theta d\theta\right]$$ Integrate each term: $$A = \frac{1}{2}\left[ 4\theta - 16(-\cos\theta) + \int_0^{\pi} 16\sin^2\theta d\theta\right]$$ To evaluate the remaining integral, use the identity \(\sin^2\theta = \frac{1}{2}(1 - \cos(2\theta))\): $$A = \frac{1}{2}\left[ 4\theta - 16(-\cos\theta) + 16\int_0^{\pi} \frac{1}{2}(1 - \cos(2\theta))d\theta\right]$$ Integrate the last term and evaluate the integral between the limits 0 and \(\pi\): $$A = \frac{1}{2}\left[ 4\pi - 16(-\cos\pi) + 16\left(\frac{1}{2}\theta - \frac{1}{4}\sin(2\theta)\right)\Big|_0^\pi\right]$$ Plugging in the limits and simplifying, we get the final answer: $$A =\frac{1}{2}\left[4\pi+16\pi - \frac{1}{2}(2\pi)\right] = 9\pi$$ So the area of the region bounded by the limaçon \(r = 2 - 4\sin\theta\) is \(9\pi\).

Key concepts

Understanding the Limaçon in Polar Coordinates

The concept of a limaçon, which is a type of polar curve, can be initially daunting. However, think of it as a curve that looks similar to a snail shell. In polar coordinates, curves are expressed with equations relating to the radius and an angle \(\theta\). The limaçon in our given problem is defined by the equation \(r = 2 - 4\sin\theta\).

• In this case, \(r\) (the distance from the origin) varies as the angle \(\theta\) changes.
• The term "limaçon" is derived from the French word for snail, representing the shape quite well.This equation is part of a broader category of curves known as "roulette" curves.

A key characteristic of this limaçon is it sometimes forms an inner loop when the coefficient of the sine term is larger than the constant, as is the case here (with -4 being larger than 2). Understanding the range of \(\theta\) is crucial because it dictates the segment of the limaçon you will be evaluating.

Integration Over Polar Curves: The Limaçon Example

Integration is the mathematical process of finding areas under curves, among other applications. When dealing with polar equations like the limaçon, integration helps us find the area enclosed by these curves.
First, identify the appropriate bounds for \(\theta\). As per the step-by-step solution, our bounds are from \(0\) to \(\pi\). This range considers the entire upper part of the limaçon, capturing all necessary segments of the curve.

• The integral we use is set in the form \(\frac{1}{2}\int (r(\theta))^2 \ d\theta\).
• This particular form accommodates the radial symmetry and accounts for the sector of the circle traced as \(\theta\) spans its limits.

By squaring the expression for \(r\) (\(r = 2 - 4\sin\theta\)), we get \((2 - 4\sin\theta)^2\). The expansion and simplification requires handling trigonometric identities carefully, such as for \(\sin^2\theta\). This integration technique allows us to calculate the precise area encased by the curve.

Finding the Area Under the Limaçon Curve

Calculating the area under a curve like a limaçon involves several steps, often requiring integration as described. The operation provides insight into the space the curve encloses. In our example, the limaçon \(r = 2 - 4\sin\theta\) has specific methods required to find the area efficiently.

• The integration setup is \(A = \frac{1}{2}\int_0^\pi (2-4\sin\theta)^2 d\theta\).
• This setup includes squaring the polar equation and integrating within the correctly chosen bounds.

Using trigonometric identities simplifies the expression, particularly \(\sin^2\theta\), which is evaluated using \(\sin^2\theta = \frac{1}{2}(1 - \cos(2\theta))\).
This transformation aids in simplifying the integral, ensuring each part is handled methodically. Finally, solving this step-by-step produces an answer of \(9\pi\), depicting the energy efficiency of the integration process in deriving the bounded area of the limaçon.