Problem 22 - Thomas Calculus

Question

Find the lengths of the curves. The curve \(r=a \sin ^{2}(\theta / 2), \quad 0 \leq \theta \leq \pi, \quad a>0\)

Step-by-Step Solution

Answer

Apply the polar curve length formula to find the length.

Step 1: Curve Formula Identification

The curve is given in polar coordinates as \(r = a \sin^2(\theta/2)\) with \(0 \leq \theta \leq \pi\). Our task is to find the length of this curve.

Step 2: Polar Curve Length Formula

In polar coordinates, the length \(L\) of a curve \(r = f(\theta)\) from \(\theta = \alpha\) to \(\theta = \beta\) is given by the formula \(L = \int_{\alpha}^{\beta} \sqrt{\left(\frac{dr}{d\theta}\right)^2 + r^2} \, d\theta\).

Key Concepts

Curve Length

When we deal with curves in mathematics, one of the fascinating tasks is finding their lengths. Curve length helps us measure the distance along a path or the boundary of a shape. In the realm of calculus, curve length is an important application that arises in different coordinate systems. It plays a vital role when interpreting abstract mathematical symbols into practical scenarios. Knowing how to calculate it enhances your understanding of geometric properties of shapes and paths.

The length of a curve can be thought of as unfurling the curve into a straight line and measuring it. Mathematical tools allow us to do this through calculus, integrating small changes along the curve to find the total length. Understanding curve length is crucial, especially in fields like physics, engineering, and computer graphics where accurate measurements are key.

Polar Curve Length Formula

Polar coordinates offer a different way to explore and solve mathematical problems traditional Cartesian coordinates might not handle as easily. When dealing with polar coordinates, the concept of finding a curve's length includes using specific formulas designed for this system.

In polar coordinates, the length of a curve expressed as \( r = f(\theta) \) is calculated using a special integral formula:

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

This formula accounts for the nature of polar curves, which involve both the radius and its change concerning the angle \( \theta \). The formula integrates over the interval from \( \alpha \) to \( \beta \), providing an accurate length calculation for curves with no straightforward straight-line equivalent.

By utilizing this formula, it's possible to calculate complex shapes' lengths seamlessly. This comes in handy particularly in advanced engineering, astronomy, and physics, where modeling and analyzing curves about different poles and origins are necessary.

Integration in Polar Coordinates

Integration in polar coordinates is essential for handling intricate shapes and curves. Unlike Cartesian coordinates, polar coordinates use radius \( r \) and angle \( \theta \) to define points, offering a unique perspective and method for solving integrals.

To integrate a function in polar coordinates, we typically replace the standard \( x \) and \( y \) coordinates with expressions involving \( r \) and \( \theta \). This often involves transforming an initially complex integral into a simpler one. The key lies in understanding and applying the correct transformations and formulas. For example, the integral we use for curve length in polar coordinates requires calculating both \( r \) and its derivative \( \frac{dr}{d\theta} \). The integration then proceeds by squaring these, adding them, and taking the square root before proceeding with the integration over the specified interval.

Mastering integration in polar coordinates enhances problem-solving skills, making it easier to address problems involving circular, elliptical, or other non-linear shapes. It broadens mathematical application possibilities, proving essential in scientific computations and various technological fields.