Problem 29 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the total area of the rose \(r=a \cos n \theta\), where \(n\) is a positive integer.

Step-by-Step Solution

Answer

The total area of the rose is \( \frac{a^2 \pi}{4} \).

Step 1: Understanding the Exercise

The problem asks us to find the total area of a rose curve given by the polar equation \( r = a \cos n\theta \). This type of curve has petals, where the number of petals depends on whether \( n \) is even or odd. Specifically, if \( n \) is odd, the rose will have \( n \) petals, and if \( n \) is even, it will have \( 2n \) petals.

Step 2: Setup the Polar Area Formula

In polar coordinates, the area \( A \) enclosed by a curve is given by \( A = \frac{1}{2} \int r^2 d\theta \). For the rose curve, substitute \( r = a \cos n\theta \) into this formula, yielding \( A = \frac{1}{2} \int (a \cos n\theta)^2 d\theta \).

Step 3: Simplify the Integral

Substitute \( r = a \cos n\theta \) into the square, giving \( A = \frac{1}{2} \int a^2 \cos^2 (n\theta) d\theta \). This simplifies to \( A = \frac{a^2}{2} \int \, \cos^2 (n\theta) d\theta \). Use the trigonometric identity \( \cos^2x = \frac{1 + \cos 2x}{2} \) to rewrite the integral as \( A = \frac{a^2}{4} \int (1 + \cos 2n\theta) \, d\theta \).

Step 4: Integrate the Expression

Now, integrate each term separately. The integral of \( 1 \) with respect to \( \theta \) is \( \theta \), and the integral of \( \cos 2n\theta \) is \( \frac{1}{2n} \sin 2n\theta \). This gives \( A = \frac{a^2}{4} \left[ \theta + \frac{1}{2n} \sin 2n\theta \right] \).

Step 5: Evaluate Over the Relevant Interval

The limits for \( \theta \) depend on whether \( n \) is even or odd. For a single petal, the curve is traced over an interval of \( \frac{\pi}{n} \) for odd \( n \), so \( \theta \) ranges from 0 to \( \frac{\pi}{n} \). Multiply the area by the number of petals, \( n \), resulting in \( A_{total} = n \times \frac{a^2}{4} \left[ \frac{\pi}{n} \right] = \frac{a^2 \pi}{4} \). For even \( n \), cover a petal over \( \frac{\pi}{n} \) and sum over the \( 2n \) petals, yielding the same total area formula.

Key Concepts

Rose Curve

A rose curve is an interesting pattern in polar coordinates described by the equation \( r = a \cos n\theta \) or \( r = a \sin n\theta \). These curves are called rose curves because their shapes resemble rose petals.
Whether the curve has petals and how many depends on the integer \( n \):

• If \( n \) is odd, the rose curve displays exactly \( n \) petals.
• If \( n \) is even, the curve has \( 2n \) petals because each petal counts double.

The parameter \( a \) influences the size of the petals. Larger values of \( a \) result in larger petals, making the rose more spread out on the polar plot.
A rose curve's elegant symmetry makes it a fascinating subject for exploration in polar coordinates and contributes to various applications in physics and engineering.

Area in Polar Coordinates

When calculating the area of a curve described in polar coordinates, we use a specialized formula. This is due to how space is represented in the polar coordinate system. To find the area, we use:

• The formula \( A = \frac{1}{2} \int r^2 d\theta \).
• Substitute the expression for \( r \) from the polar equation into this formula, and simplify.

For the rose curve \( r = a \cos n\theta \), the integral for the area becomes:\[A = \frac{1}{2} \int (a \cos n\theta)^2 d\theta = \frac{a^2}{2} \int \cos^2 (n\theta) d\theta.\]To further simplify, we apply trigonometric identities, allowing easier integration to find the area each petal encompasses.
This process allows determining both the area of individual petals and the total area by considering all petals.

Trigonometric Identities

Trigonometric identities are crucial tools that help simplify complex trigonometric expressions. In polar coordinates, we encounter these identities to ease the integration process of polar functions like the rose curve.

• One key identity used here is \( \cos^2 x = \frac{1 + \cos 2x}{2} \).
• This identity transforms the integrand into a more manageable form, breaking it down into components that are easier to integrate.

By adopting this identity, the area integral becomes:\[A = \frac{a^2}{4} \int (1 + \cos 2n\theta) d\theta.\]This aids in simplifying and solving the integral as:

• The integral of \( 1 \) becomes \( \theta \) over the interval.
• The integral of \( \cos 2n\theta \) turns into \( \frac{1}{2n} \sin 2n\theta \).

Thus, trigonometric identities not only facilitate solving these integrals but also deepen our understanding of mathematical symmetries and their aesthetic beauty.