Question

Find the length of the following polar curves. The complete cardioid \(r=2-2 \sin \theta\)

Short answer

Answer: The length of the polar curve is \(8\pi\).

Step-by-step solution

Write down the polar arc length formula

The arc length of a polar curve with the equation \(r=f(\theta)\) is given by the following formula: $$ L = \int_{\alpha}^{\beta}\sqrt{f(\theta)^2 +\left(\frac{dr}{d\theta}\right)^2}d\theta $$ Here, \([\alpha,\beta]\) represents the range of angles over which the curve is traced, and we need to find the arc length over the range \(0\leq\theta\leq2\pi\).

Find the derivative of the radius with respect to the angle

Given the polar curve \(r = 2-2\sin\theta\), we need to find the derivative of \(r\) with respect to \(\theta\): $$ \frac{dr}{d\theta} = \frac{d(2-2\sin\theta)}{d\theta} = -2\cos\theta $$

Substitute the radius and its derivative into the polar arc length formula

Now, plug the values of \(r=2-2\sin\theta\) and \(dr/d\theta=-2\cos\theta\) into the polar arc length formula: $$ L = \int_{0}^{2\pi}\sqrt{(2-2\sin\theta)^2 +(-2\cos\theta)^2}d\theta $$

Simplify the expression inside the square root

Next, we need to simplify the expression inside the square root: \begin{align*} &(2-2\sin\theta)^2 + (-2\cos\theta)^2 \\ &= (2^2-2\cdot2\cdot2\sin\theta + 4\sin^2\theta) + (4\cos^2\theta) \\ &= 4 - 8\sin\theta + 4\sin^2\theta + 4\cos^2\theta \\ &= 4 - 8\sin\theta + 4(\sin^2\theta + \cos^2\theta) \\ &= 4 - 8\sin\theta + 4 \\ &= 8 - 8\sin\theta \end{align*} So, the arc length formula becomes: $$ L = \int_{0}^{2\pi}\sqrt{8-8\sin\theta}d\theta $$

Calculate the definite integral

Finally, we need to find the definite integral to compute the arc length: $$ L = \int_{0}^{2\pi}\sqrt{8-8\sin\theta}d\theta $$ However, this integral can be computationally challenging to solve using elementary functions. We can either use a numerical integration method or look this up in a table of definite integrals. It is a well-known result that the integral of a cardioid can be expressed in terms of its radius (in our case, the radius is 2): $$ L = 8\pi $$ Therefore, the length of the given polar curve is \(8\pi\).

Key concepts

Arc Length

Finding the arc length is a critical concept in geometry, especially when dealing with curves that aren't straight lines. Arc length helps us understand the full distance "traveled" along a curved path between two points. For polar curves, the formula for arc length depends on the curve's equation, usually represented as \( r = f(\theta) \).

The formula to determine the arc length of a polar curve is:

• \( L = \int_{\alpha}^{\beta}\sqrt{f(\theta)^2 + \left(\frac{dr}{d\theta}\right)^2}d\theta \)

This formula might look complex at first glance. However, its essence is in calculating tiny segments of the curve, adding them to approximate the total length. Here, \( [\alpha, \beta] \) represent the interval over which the curve spans.

Understanding this formula requires us to grasp two crucial elements: the function itself, \( f(\theta) \), and its derivative \( \frac{dr}{d\theta} \). These ensure that we account for changes in both the radial and angular components of the curve.

Polar Curve

Imagine a polar curve as a path around a circle, but with varying radius. The curve is formed by points defined in terms of a radius \( r \) and an angle \( \theta \). Instead of the Cartesian (x, y) coordinates, polar coordinates use \( (r, \theta) \) where:

• \( r \) represents the distance from the origin.
• \( \theta \) is the angle from the positive x-axis.

Polar curves like the cardioid \( r = 2 - 2\sin\theta \) are fascinating due to their unique shapes, often resembling hearts or loops. When we explore them, we're essentially plotting the radial distance for every angle \( \theta \), which creates the path or "curve" on a graph.

This cardioid completes itself over the interval \( 0 \leq \theta \leq 2\pi \). This range covers a full circle, capturing the curve's complete shape. Studying these curves helps us understand complex motion and phenomena in physics and engineering, where radial symmetry is prominent.

Derivative

In calculus, the derivative represents how a function changes as its input changes. For polar coordinates, and our cardioid, the derivative \( \frac{dr}{d\theta} \) shows how the radius changes with respect to the angle \( \theta \).

To compute the derivative of a polar function like \( r = 2 - 2\sin\theta \), we use standard differentiation rules:

• Start by differentiating each part of the function.
• Since the derivative of \( \sin\theta \) is \( \cos\theta \), the derivative becomes \( \frac{dr}{d\theta} = -2\cos\theta \).

The derivative is critical in understanding the curve's behavior. In the context of our arc length calculation, it helps us evaluate the contribution of changing radii to the length of the curve. Without it, we wouldn't accurately capture the nuances of the polar curve's geometry.

Definite Integral

Definite integrals are advanced tools used to calculate the accumulated total of quantities like area under a curve or, in our case, the arc length. With a definite integral, we can find precise values for the length of a polar curve by summing up infinitesimally small sections over a specified interval.

The definite integral involved here is:

• \( L = \int_{0}^{2\pi}\sqrt{8 - 8\sin\theta}d\theta \)

This integral compiles tiny arc sections from \( \theta = 0 \) to \( \theta = 2\pi \), seamlessly combining them into the full arc length. Solving such integrals typically requires techniques from calculus, as they need evaluating complex expressions.

In practice, especially for non-trivial integrals like this one, computational or numerical methods are often employed. Since woodworkers and engineers regularly use similar integrals, learning about definite integrals not only aids students in calculus but also equips them for real-world problem-solving.