Question

Find the total arc length of \(r=3 \sin \theta\).

Short answer

The total arc length is \(3\pi\).

Step-by-step solution

Convert to Cartesian Coordinates

The given polar equation is \( r = 3 \sin \theta \). Recall that in polar coordinates, \( x = r \cos \theta \) and \( y = r \sin \theta \). Substituting \( y = r \sin \theta = 3 \sin^2 \theta \) and \( x = r \cos \theta = 3 \sin \theta \cos \theta \). This converts the equation to Cartesian; however, for arc length, it might be sufficient to work in polar form.

Use Polar Arc Length Formula

The arc length \( L \) of a curve defined by \( r = f(\theta) \) from \( \theta = a \) to \( \theta = b \) is given by the integral \( L = \int_a^b \sqrt{ \left( \frac{dr}{d\theta} \right)^2 + r^2 } \, d\theta \).

Determine the Integral Limits

The polar function \( r = 3 \sin \theta \) represents a circle. For a complete circle in polar coordinates where \( r = a \sin \theta \), the range of \( \theta \) is from \( 0 \) to \( \pi \).

Differentiate r with Respect to Theta

Calculate the derivative \( \frac{dr}{d\theta} \). Since \( r = 3 \sin \theta \), we have \( \frac{dr}{d\theta} = 3 \cos \theta \).

Set up the Integral

Substitute \( r \) and \( \frac{dr}{d\theta} \) into the arc length formula: \[ L = \int_0^\pi \sqrt{ (3 \cos \theta)^2 + (3 \sin \theta)^2 } \, d\theta = \int_0^\pi \sqrt{ 9 \cos^2 \theta + 9 \sin^2 \theta } \, d\theta \]

Simplify the Integral

Using the trigonometric identity \( \cos^2 \theta + \sin^2 \theta = 1 \), the integral simplifies to \[ L = \int_0^\pi \sqrt{9 (\cos^2 \theta + \sin^2 \theta)} \, d\theta = \int_0^\pi \sqrt{9} \, d\theta = \int_0^\pi 3 \, d\theta \].

Evaluate the Integral

Now evaluate the integral: \[ L = 3 \int_0^\pi \, d\theta = 3[\theta]_0^\pi = 3(\pi - 0) = 3\pi \].

Key concepts

Polar Coordinates

Polar coordinates are an alternative to Cartesian coordinates for describing the location of a point in the plane. Instead of using a horizontal and vertical measure (x, y), polar coordinates use a distance from the origin, called the radius (r), and an angle (θ). These are particularly useful in problems with symmetry or where curves are naturally circular, as they can simplify calculations and reveal a clearer geometric understanding.

When converting between polar and Cartesian coordinates, it's essential to use the relations:

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

It's fascinating how equations can shift form based on the coordinate system. For instance, the polar equation \( r = 3 \sin \theta \) can describe a circle, which can seem more complex in Cartesian coordinates. Understanding these conversions aids in interpreting the nature and symmetry of curves effectively.

Calculus Integration

Calculus integration is integral to finding the arc length of curves in polar coordinates. Unlike derivatives that measure the rate of change, integration accumulates quantities over a range, such as adding up infinitesimal pieces of a curve.

The formula for arc length in polar coordinates is:
\[ L = \int_a^b \sqrt{\left( \frac{dr}{d\theta} \right)^2 + r^2} \, d\theta \]
This formula integrates from \(\theta = a\) to \(\theta = b \), calculating the total length of the curve by summing these infinitesimal lengths of the arc. This cumulative approach allows calculus to solve complex geometry problems that lack straightforward solutions.

Trigonometric Identities

Trigonometric identities simplify many calculus problems involving polar coordinates. The most essential identity used is \( \sin^2 \theta + \cos^2 \theta = 1 \). This identity helps in reducing complicated expressions during integration.

In our specific example, this identity allowed us to simplify the integrand in the formula for arc length as:
\[ \sqrt{9 \cos^2 \theta + 9 \sin^2 \theta} = \sqrt{9(\cos^2 \theta + \sin^2 \theta)} \]
Using the identity, it reduces to \( \sqrt{9} = 3 \), thus simplifying the evaluation of the integral and saving a great deal of computational effort.

Derivative of Polar Functions

In calculus, the derivative of a function describes the rate of change of the function's output value concerning changes in input value. For polar coordinates, determining the derivative \( \frac{dr}{d\theta} \) is crucial for calculating arc length.

Given \( r = 3 \sin \theta \), differentiating with respect to \( \theta \) gives:
\( \frac{dr}{d\theta} = 3 \cos \theta \)
This derivative measures how the radius changes as the angle \( \theta \) varies. By including this in our arc length formula, we account for the variation along the curve, ensuring our integration captures the entire length accurately.

Such derivatives are foundational in calculus and serve as building blocks for analyzing more complex curves and dynamics.