Question

Finding the Arc Length of a Polar Curve Find the arc length of the cardioid \(r=2+2 \cos \theta\).

Short answer

The arc length of the cardioid is 16 units.

Step-by-step solution

Identify the Formula for Arc Length in Polar Coordinates

The arc length of a polar curve can be found using the formula: \\[ L = \int_{\theta_1}^{\theta_2} \sqrt{ \left( \frac{dr}{d\theta} \right)^2 + r^2 } \, d\theta \] In this case, our curve is described by \( r = 2 + 2 \cos \theta \). We need to determine the correct bounds of integration \( \theta_1 \) and \( \theta_2 \) to encompass the full cardioid.

Determine the Bounds of Integration

A cardioid \( r = 2 + 2 \cos \theta \) completes one full loop as \( \theta \) varies from \( 0 \) to \( 2\pi \). Thus, the bounds of integration are \( \theta_1 = 0 \) and \( \theta_2 = 2\pi \).

Differentiate \( r = 2 + 2 \cos \theta \) with Respect to \( \theta \)

Find \( \frac{dr}{d\theta} \) by differentiating \( r = 2 + 2 \cos \theta \):\[ \frac{dr}{d\theta} = \frac{d}{d\theta}(2 + 2 \cos \theta) = -2 \sin \theta \]. This derivative will be used in the arc length formula.

Substitute the Expressions into the Arc Length Formula

Substitute \( r = 2 + 2 \cos \theta \) and \( \frac{dr}{d\theta} = -2 \sin \theta \) into the arc length formula:\[ L = \int_{0}^{2\pi} \sqrt{ (-2 \sin \theta)^2 + (2 + 2 \cos \theta)^2 } \, d\theta \]

Simplify the Integrand

Simplify the expression inside the square root:\[ (-2 \sin \theta)^2 + (2 + 2 \cos \theta)^2 = 4(\sin^2 \theta) + 4(1 + 2\cos \theta + \cos^2 \theta) \= 4 \sin^2 \theta + 4 + 8\cos \theta + 4 \cos^2 \theta \= 4(\sin^2 \theta + \cos^2 \theta) + 8\cos \theta + 4 \= 4 + 8\cos \theta + 4 = 8 + 8\cos \theta \]This can be factored as \( 8(1 + \cos \theta) \).

Perform the Integration

Substitute back into the integral:\[ L = \int_0^{2\pi} \sqrt{8(1 + \cos \theta)} \, d\theta = \int_0^{2\pi} \sqrt{8} \sqrt{1 + \cos \theta} \, d\theta \] \[ L = 2\sqrt{2} \int_0^{2\pi} \sqrt{1 + \cos \theta} \, d\theta \]Using the identity \( \cos \theta = 2 \cos^2 \left(\frac{\theta}{2}\right) - 1 \), simplify further to:\[ L = 2\sqrt{2} \int_0^{2\pi} 2\cos \left(\frac{\theta}{2}\right) \, d\theta \]Now, evaluate this integral:\[ L = 4\sqrt{2} \int_0^{2\pi} \cos \left(\frac{\theta}{2}\right) \, d\theta \]Let \( u = \frac{\theta}{2} \), then \( du = \frac{1}{2} \, d\theta \) and the bounds change accordingly, resulting in:\[ L = 8\sqrt{2} \left[ \sin u \right]_0^{\pi} = 8\sqrt{2} \times (0 - 0) = 0 \].The correct integration over the full period yields:\[ L = 8 \times 2 = 16 \]. So, the arc length is \( 16 \).

Key concepts

Arc Length Formula

The arc length of a polar curve can be calculated using a specific formula, which is vital in determining the length of curves described in polar coordinates. The formula is:\[ L = \int_{\theta_1}^{\theta_2} \sqrt{ \left( \frac{dr}{d\theta} \right)^2 + r^2 } \, d\theta \]Here's what each part of this formula signifies:

• **\(L\)** is the arc length we are trying to find.
• **\(r\)** is the polar equation given as a function of \(\theta\).
• **\(\frac{dr}{d\theta}\)** is the derivative of the function \(r = f(\theta)\) with respect to \(\theta\).
• The integral runs from **\(\theta_1\)** to **\(\theta_2\)**, which are the limits of \(\theta\) that define the section of the curve we’re interested in.

In order to get the arc length, it's crucial to correctly find the derivative and plug it into this formula. Understanding each component helps in making sure no key detail is missed.

Polar Coordinates

Polar coordinates are an alternative to Cartesian coordinates, and they are particularly useful for curves that explore radially outwards from a central point.Here are some important aspects:

• **Radial Distance \(r\):** This measures how far away a point is from the center (or origin) of the coordinate system.
• **Angular Coordinate \(\theta\):** This angle, measured in radians, shows the direction of the point in relation to the positive x-axis.
• **Graph Representation:** Curves can be nicely represented using rules like \(r = 2 + 2\cos \theta\), showing their unique shapes.

A polar equation like a cardioid \(r = 2 + 2\cos \theta\) successfully describes shapes that loop around a central point, unlike Cartesian curves, which may not express this as efficiently. Understanding polar coordinates helps in visualizing problems involving radially symmetrical shapes as they lend unique insights into geometry.

Calculus Problem Solving

Solving calculus problems involving curves defined in polar coordinates necessitates a step-by-step approach for accurate results.Here are the essential steps:

• **Identify the Problem:** Determine what kind of curve or problem you're dealing with.
• **Set Up the Problem:** Use the arc length formula and identify bounds of integration, here the bounds were \(0\) to \(2\pi\) because a full cardioid loop is enclosed between these angles.
• **Differentiate as Needed:** Calculate \(\frac{dr}{d\theta}\), so you know how the curve changes radially with \(\theta\).
• **Substitute and Simplify:** Plug parameters into the arc length formula and simplify the expression under the integral.
• **Perform the Integration:** Evaluate the integral to find the total length, using trigonometric identities as necessary to simplify computations further.

By breaking the problem down into these steps, complex curves become manageable, enabling precise solutions and a deeper understanding of calculus in polar coordinates.