Problem 26 - Calculus Early Transcendentals: Pearson New International Edition

Question

Let \(r=f(\theta)\), where \(f\) is continuous on the closed interval \([\alpha, \beta]\). Derive the following formula for the length \(L\) of the corresponding polar curve from \(\theta=\alpha\) to \(\theta=\beta\). $$ L=\int_{\alpha}^{\beta} \sqrt{[f(\theta)]^{2}+\left[f^{\prime}(\theta)\right]^{2}} d \theta $$

Step-by-Step Solution

Answer

The formula is derived by finding derivatives and using the arc length formula.

Step 1: Understand the problem

To solve the problem of finding the length \(L\) of the polar curve \(r = f(\theta)\) from \(\theta = \alpha\) to \(\theta = \beta\), we need to derive a formula using calculus. The goal is to derive the integral expression provided for the arc length of the curve.

Step 2: Identify the polar coordinates

In polar coordinates, the position of a point on the curve is given by \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\). Here, \(r = f(\theta)\). We need expressions for both \(x\) and \(y\) in terms of \(\theta\).

Step 3: Compute the derivatives

Find the derivatives \(\frac{dx}{d\theta}\) and \(\frac{dy}{d\theta}\). Using \(x = f(\theta) \cos(\theta)\) and \(y = f(\theta) \sin(\theta)\), apply the product rule: - \(\frac{dx}{d\theta} = f'(\theta) \cos(\theta) - f(\theta) \sin(\theta)\) - \(\frac{dy}{d\theta} = f'(\theta) \sin(\theta) + f(\theta) \cos(\theta)\).

Step 4: Use the arc length formula

The general formula for arc length in Cartesian coordinates is \(L = \int \sqrt{\left(\frac{dx}{d\theta}\right)^2 + \left(\frac{dy}{d\theta}\right)^2} \, d\theta\). Substitute the derivatives calculated in Step 3 into this expression.

Step 5: Simplify the expression

Substitute: \[ \left(\frac{dx}{d\theta}\right)^2 = \left(f'(\theta) \cos(\theta) - f(\theta) \sin(\theta)\right)^2 \]\[ \left(\frac{dy}{d\theta}\right)^2 = \left(f'(\theta) \sin(\theta) + f(\theta) \cos(\theta)\right)^2 \]Add these and simplify using trigonometric identities to find:\[ \left[f(\theta)\right]^2 + \left[f'(\theta)\right]^2 \] after simplification.

Step 6: Derive the integral

Thus, the integral representing the arc length becomes: \[ L = \int_{\alpha}^{\beta} \sqrt{[f(\theta)]^2 + \left[f'(\theta)\right]^2} \, d\theta \]This matches the expression provided in the problem statement. The derivation is complete.

Key Concepts

Arc Length Formula

The arc length formula is a powerful tool used in calculus to determine the length of a curve. In the context of polar coordinates, where curves are defined by equations of the form \(r = f(\theta)\), the arc length formula helps measure the 'distance' along the curve from one point to another in terms of angle. The key expression we derived is:
\[L = \int_{\alpha}^{\beta} \sqrt{[f(\theta)]^{2} + \left[f^{\prime}(\theta)\right]^{2}} \, d\theta\]
This formula tells us that to find the length \(L\) of a polar curve from \(\theta = \alpha\) to \(\theta = \beta\), we integrate the square root of the sum of the squares of two terms, \([f(\theta)]^{2}\) and \([f'(\theta)]^{2}\), over the interval. Each part of the formula captures different components of the curve's geometry, and combined, they provide a holistic measure of length.

Polar Coordinates

Polar coordinates offer a unique way to represent points on a plane using a radius and an angle rather than the traditional Cartesian \((x, y)\) coordinates. This system is especially useful for curves that are naturally circular or spiral in shape.
Within this system, any point is described using \(r\), the distance from the origin, and \(\theta\), the angle measured from the positive x-axis. For a curve defined by \(r = f(\theta)\), \(r\) varies as \(\theta\) changes, tracing the curve as \(\theta\) goes from \(\alpha\) to \(\beta\).
Helpful formulas in this coordinate system include:

• \(x = r \cos(\theta)\)
• \(y = r \sin(\theta)\)

These transformations allow us to convert between polar and Cartesian systems, which is essential in deriving the arc length formula.

Calculus Derivation

Deriving the arc length formula for polar coordinates requires calculus techniques due to the complex interrelationship between \(r\), \(\theta\), and the Cartesian coordinates. The derivation hinges on expressing the polar curve in terms of \(x\) and \(y\), then relating these back to \(\theta\).
We begin by translating \(r = f(\theta)\) into the Cartesian format:

• \(x = f(\theta) \cos(\theta)\)
• \(y = f(\theta) \sin(\theta)\)

From here, we need the derivatives \(\frac{dx}{d\theta}\) and \(\frac{dy}{d\theta}\), because the general arc length formula in its Cartesian form is:
\[L = \int \sqrt{\left(\frac{dx}{d\theta}\right)^2 + \left(\frac{dy}{d\theta}\right)^2} \, d\theta\]
This approach allows us to find precise changes along \(x\) and \(y\) as \(\theta\) changes, laying the groundwork to evaluate the integral representing the curve's length.

Product Rule

The product rule is a fundamental calculus rule used to differentiate expressions involving products of functions. It states that if you have two functions, \(u(\theta)\) and \(v(\theta)\), the derivative of their product is:
\[\frac{d}{d\theta}[u(\theta)v(\theta)] = u'(\theta)v(\theta) + u(\theta)v'(\theta)\]
In the context of deriving the polar arc length, the product rule is crucial when differentiating:

• \(x = f(\theta) \cos(\theta)\)
• \(y = f(\theta) \sin(\theta)\)

Applying the product rule, we obtain:

• \(\frac{dx}{d\theta} = f'(\theta) \cos(\theta) - f(\theta) \sin(\theta)\)
• \(\frac{dy}{d\theta} = f'(\theta) \sin(\theta) + f(\theta) \cos(\theta)\)

These derivatives are essential for substituting into the arc length formula to transition from polar to Cartesian coordinates and ultimately simplify the calculation of the curve's length.