Problem 8 - Essential Calculus

Question

Find the area of the region. One petal of \(r=\cos 5 \theta\)

Step-by-Step Solution

Answer

The area of one petal of the polar graph \(r=\cos 5 \theta\) is \(\frac{\pi}{20}\).

Step 1 - Identify the Limits of Integration

To find the limits of integration, we need to find where \(r=0\) for \(r=\cos 5\theta\), that is, where \(\cos 5\theta = 0\). Solving this equation gives us \(\theta = \frac{\pi}{10}\) and \(\theta = \frac{3\pi}{10}\). These limits define a full petal in the polar graph.

Step 2 - Integrate to Find the Area

The area within a polar graph between angles \(\alpha\) and \(\beta\) is calculated by \(\frac{1}{2} \int_\alpha^\beta r^2 d\theta\). In this case, \(r=\cos 5\theta\) so the integral becomes \(\frac{1}{2} \int_{\pi/10}^{3\pi/10} (\cos 5\theta)^2 d\theta\). We can simplify the equation by using the double angle identity for cosine - \(\cos^2(n\theta) = 0.5(1+\cos(2n\theta)\). Applying this identity yields the integral becomes \(\frac{1}{2} \int_{\pi/10}^{3\pi/10} 0.5(1+\cos(10\theta)) d\theta\). However, since \(\int_{\pi/10}^{3\pi/10} \cos(10\theta) d\theta =0\), this simplifies to 0.25 times \(\int_{\pi/10}^{3\pi/10} d\theta\), which equals 0.25 times \[\beta - \alpha = \frac{3\pi}{10} - \frac{\pi}{10}\].

Step 3 - Calculate the Integral

Solving the resulting equation: 0.25 times \(\frac{2\pi}{10} = \frac{\pi}{20}\). Thus, the area of one petal in the polar graph of \(r=\cos 5\theta\) is \(\frac{\pi}{20}\).

Key Concepts

Limits of Integration

Understanding the limits of integration is crucial when working with polar coordinates. These limits, essentially, define the region over which we are integrating. For instance, when determining the area of a petal described by the polar equation r = \( \cos 5\theta \), we begin by identifying where r equals zero. This corresponds to the angles where \cos(5\theta) = 0, and solving this provides us with the necessary boundaries for integration. Identifying the right limits ensures that we calculate the area of only one petal. In the exercise, the limits are \( \frac{\pi}{10} \) and \( \frac{3\pi}{10} \), which bounds a single petal on the polar graph. These values are paramount in setting up the integral for the area calculation.

Double Angle Identity

The double angle identity is an important trigonometric identity that simplifies complex expressions and makes integration manageable. In the context of polar graph area calculation, the identity transforms \( \cos^2(5\theta) \) by using the formula \( \cos^2(n\theta) = 0.5[1 + \cos(2n\theta)] \). Applying this identity simplifies the integrand, allowing for a straightforward integration. Without such identities, computing the area directly from the original polar equation would be considerably more complicated.

Polar Graph Integration

Performing integration on a polar graph requires a unique approach where the area element is expressed as \( \frac{1}{2} r^2 d\theta \). The half in the equation accounts for the fact that polar coordinates measure area in a circular pattern, rather than in rectangles as with Cartesian coordinates. The integral \( \int_\alpha^\beta \frac{1}{2} r^2 d\theta \) gauges the area from angle \alpha to angle \beta with a function r(\theta). In the given problem, by integrating \( \cos 5\theta \) squared over the identified limits, we can find the area of the petal.

Polar Equations

Polar equations such as r = \( \cos 5\theta \) represent curves on the polar coordinate system, where r is the distance from the origin (pole) and \theta is the angle from the polar axis. Each value of \theta corresponds to a point on the curve, and as \theta varies, r changes accordingly, tracing the shape of the curve. In the case of \( r = \cos 5\theta \), the equation describes a rose curve with five petals. Each petal can be isolated and analyzed individually by utilizing the correct limits of integration as we did in this exercise.