Question

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

Short answer

The total area is \( \frac{a^2 \pi}{4} \) if \(n\) is odd and \( \frac{a^2 \pi}{2} \) if \(n\) is even.

Step-by-step solution

Understanding the Problem

We need to find the total area enclosed by the rose curve given by the polar equation \( r = a \cos(n\theta) \). Here, \(a\) and \(n\) are constants, with \(n\) being a positive integer.

Formula for Area in Polar Coordinates

The area \( A \) enclosed by a polar curve \( r = f(\theta) \) from \( \theta = \alpha \) to \( \theta = \beta \) is given by:\[A = \frac{1}{2} \int_{\alpha}^{\beta} r^2 \, d\theta\]For our curve, \( r = a \cos(n\theta) \), so we substitute this into the formula.

Determine Integration Limits

The rose curve \( r = a \cos(n\theta) \) completes one full petal when \( \theta \) ranges from \( 0 \) to \( \frac{\pi}{n} \). Since the number of petals when \( n \) is odd is \( n \), we need to consider the range from \( 0 \) to \( 2\pi \) for the total area. If \( n \) is even, there are \( 2n \) petals but symmetry still simplifies the limits to \( \theta \) from \( 0 \) to \( \pi \).

Calculate Integral for One Petal

Calculate the area for one petal by evaluating the integral:\[A = \frac{1}{2} \int_{0}^{\frac{\pi}{n}} [a \cos(n\theta)]^2 \, d\theta = \frac{1}{2} \int_{0}^{\frac{\pi}{n}} a^2 \cos^2(n\theta) \, d\theta\]

Use Trigonometric Identity to Simplify Integral

We use the identity \( \cos^2(\theta) = \frac{1 + \cos(2\theta)}{2} \) to simplify the integral:\[A = \frac{1}{2} \int_{0}^{\frac{\pi}{n}} a^2 \left( \frac{1 + \cos(2n\theta)}{2} \right) \, d\theta = \frac{a^2}{4} \int_{0}^{\frac{\pi}{n}} (1 + \cos(2n\theta)) \, d\theta\]

Evaluate Integral

Split the integral and evaluate each term:\[A = \frac{a^2}{4} \left[ \int_{0}^{\frac{\pi}{n}} 1 \, d\theta + \int_{0}^{\frac{\pi}{n}} \cos(2n\theta) \, d\theta \right]\]The integral of \(1\) is straightforward: \( \frac{\pi}{n} \). The integral of \( \cos(2n\theta) \) evaluates to zero over this symmetric interval. Thus,\[A = \frac{a^2}{4} \cdot \frac{\pi}{n} = \frac{a^2 \pi}{4n} \]

Total Area for All Petals

If \(n\) is odd, multiply by \(n\) to account for all petals:\[\text{Total Area} = n \cdot \frac{a^2 \pi}{4n} = \frac{a^2 \pi}{4}\]If \(n\) is even, multiply by \(2n\):\[\text{Total Area} = 2n \cdot \frac{a^2 \pi}{4n} = \frac{a^2 \pi}{2}\]

Conclusion

The total area of the rose curve \( r = a \cos n\theta \) is \( \frac{a^2 \pi}{4} \) when \(n\) is odd, and \( \frac{a^2 \pi}{2} \) when \(n\) is even.

Key concepts

rose curve area

Understanding the area of a rose curve is a fascinating topic in polar coordinates. A rose curve is a special type of polar plot that forms petal-like shapes, given by the equation \( r = a \cos(n\theta) \) for this exercise. The number of petals depends on whether \( n \) is even or odd.

To determine the total area enclosed by a rose curve, we need to sum up the area of all its petals. This calculation involves determining the integral of the curve over a specific range of \( \theta \).

For odd \( n \), the number of petals equals \( n \), and for even \( n \), the number of petals is doubled to \( 2n \). Thus, the range for the integral varies based on whether \( n \) is even or odd. The resulting area for odd \( n \) is \( \frac{a^2 \pi}{4} \), while for even \( n \), it is \( \frac{a^2 \pi}{2} \). These formulas capture the symmetry and structure inherent to rose curves.

trigonometric identity

Trigonometric identities play a crucial role in simplifying integrals involving trigonometric functions. In our rose curve exercise, the identity \( \cos^2(\theta) = \frac{1 + \cos(2\theta)}{2} \) is especially handy. It allows us to express \( \cos^2 \) in terms of a sum, a powerful technique for integration.

By substituting this identity into the integral, we transform a potentially complex integration into manageable parts involving simpler trigonometric and constant terms. This simplification step is key to executing calculations effectively.

Applying this identity not only makes the mathematics more straightforward but also illustrates the beauty and utility of trigonometric identities in solving real-world problems involving periodic functions, like those seen in rose curves and other advanced calculus scenarios.

integration in polar coordinates

Integration in polar coordinates opens new dimensions in calculus, providing a method to find areas and volumes where traditional Cartesian coordinates might be cumbersome. The integral formula for area enclosed by a curve \( r = f(\theta) \) is \[A = \frac{1}{2} \int_{\alpha}^{\beta} r^2 \, d\theta\] This formula is fundamental when dealing with functions expressed in polar coordinates.

In the rose curve, replacing \( r \) with \( a \cos(n\theta) \) leads us to integrate \( a^2 \cos^2(n\theta) \) over the appropriate interval, revealing the area pattern. Integration in polar coordinates often involves finding suitable limits, based on geometric features, for accurate computation of enclosed areas.

Understanding these integrations not only helps with courses involving advanced calculus but also is pivotal in fields like physics and engineering, where radial and angular symmetries are common.

even and odd integers in calculus

The concepts of odd and even integers have more mathematical significance than just counting. In calculus, they influence symmetry and periodicity, especially within polar coordinates and Fourier series.

For rose curves, the integer \( n \) determines the number of petals and their arrangement. An odd \( n \) results in the same number of petals as \( n \), maximizing symmetry with respect to the polar axis. Conversely, an even \( n \) yields double the petals, contributing to more complex symmetrical structures.

Recognizing these integer effects aids in understanding how functions behave under transformations and how certain integrals can be optimized or simplified. Even and odd properties lead to results that make mathematical operations more efficient and insights into the nature of patterns formed by mathematical and natural phenomena.