Question

Let \(R_{n}\) be the region bounded by the \(n\) th turn and the \((n+1)\) st turn of the spiral \(r=e^{-\theta}\) in the first and second quadrants, for \(\theta \geq 0\) (see figure). a. Find the area \(A_{n}\) of \(R_{n}\) b. Evaluate \(\lim _{n \rightarrow \infty} A_{n}\) c. Evaluate \(\lim _{n \rightarrow \infty} A_{n+1} / A_{n}\)

Short answer

To summarize the solution: a) We found the area \(A_n\) of \(R_n\) by setting up and evaluating the integral: $$ A_n = \frac{1}{2}\int_{\theta_n}^{\theta_n + \pi} (e^{-2\theta}) d\theta $$ b) We evaluated the limit of the sequence \(A_n\) as \(n \rightarrow \infty\) and found it to be 0: $$ \lim_{n \rightarrow \infty} A_n = 0 $$ c) We evaluated the limit of the ratio \(A_{n+1} / A_n\) as \(n \rightarrow \infty\) and found it to be \(e^{-2\pi}\): $$ \lim_{n \rightarrow \infty} \frac{A_{n+1}}{A_n} = e^{-2\pi} $$

Step-by-step solution

a. Find the area \(A_n\) of \(R_n\):

To find the area of \(R_n\), we will first need to determine the limits of integration for \(\theta\). We can find the limits by finding when the spiral intersects the \(x\)-axis (or, equivalently, the real axis). The intersections can be found by setting \(r = 0\). Thus, for the \(n^{th}\) and \((n + 1)^{st}\) turns, $$ r_n = e^{-\theta_n} = 0 \\\\ r_{n+1} = e^{-\theta_{n+1}} = 0 $$ Since the exponential function is equal to 0 only at the poles, we cannot solve these equations directly. However, we can determine that the distance between two intersections along the \(x\)-axis within the first and second quadrants is equal to \(\pi\). Hence, $$ \theta_{n+1} - \theta_n = \pi $$ We can now set up the integral for the area of \(R_n\) in polar coordinates. $$ A_n = \frac{1}{2}\int_{\theta_n}^{\theta_{n+1}} (e^{-\theta})^2 d\theta = \frac{1}{2}\int_{\theta_n}^{\theta_n + \pi} (e^{-2\theta}) d\theta $$ Now, we can calculate the integral.

b. Evaluate \(\lim _{n \rightarrow \infty} A_n\)

To find the limit of the sequence \(A_n\), we will first compute the integral: $$ A_n = \frac{1}{2}\int_{\theta_n}^{\theta_n + \pi} (e^{-2\theta}) d\theta = \frac{1}{4}\left[-e^{-2\theta}\right]_{\theta_n}^{\theta_n + \pi} = \frac{e^{-2\theta_n} - e^{-2\theta_n - 2\pi}}{4} $$ Then, we can find the limit as \(n \rightarrow \infty\). $$ \lim_{n \rightarrow \infty} A_n = \lim_{n \rightarrow \infty} \frac{e^{-2\theta_n} - e^{-2\theta_n - 2\pi}}{4} = 0 $$ As \(n \rightarrow \infty\), both \(e^{-2\theta_n}\) and \(e^{-2\theta_n - 2\pi}\) approach 0, so the limit of \(A_n\) is 0.

c. Evaluate \(\lim _{n \rightarrow \infty} A_{n+1} / A_n\)

To find the limit of the ratio of consecutive terms, we will simply divide \(A_{n+1}\) by \(A_n\). $$ \frac{A_{n+1}}{A_n} = \frac{e^{-2\theta_{n+1}} - e^{-2\theta_{n+1} - 2\pi}}{e^{-2\theta_n} - e^{-2\theta_n - 2\pi}} $$ We can now find the limit as \(n \rightarrow \infty\): $$ \lim_{n \rightarrow \infty} \frac{A_{n+1}}{A_n} = \lim_{n \rightarrow \infty} \frac{e^{-2\theta_{n+1}} - e^{-2\theta_{n+1} - 2\pi}}{e^{-2\theta_n} - e^{-2\theta_n - 2\pi}} = e^{-2\pi} $$ As \(n \rightarrow \infty\), the terms \(\frac{e^{-2\theta_n}}{e^{-2\theta_{n+1}}}\) and \(\frac{e^{-2\theta_n - 2\pi}}{e^{-2\theta_{n+1} - 2\pi}}\) are both equal to \(e^{-2\pi}\). So, the limit of the ratio \(A_{n+1}/A_n\) is equal to \(e^{-2\pi}\).

Key concepts

Polar Coordinates

Polar coordinates are a system of expressing points in a plane, using a distance from a reference point and an angle from a reference direction. Unlike Cartesian coordinates, which use \(x\) and \(y\) values, polar coordinates use \(r\) and \(\theta\), where \(r\) is the radius (distance from the origin) and \(\theta\) is the angle (measured in radians from the positive x-axis).
In problems involving spirals, such as this one, we often find it easier to express the curves in polar coordinates. For example, the spiral given by \(r = e^{-\theta}\) tells us how \(r\) changes in relation to \(\theta\).
When working with areas in polar coordinates, the formula \((1/2)\int_{a}^{b} (r(\theta))^2 d\theta\) is often useful. This represents the integral needed to find the area of a sector, dictated by the curve \(r(\theta)\) from angle \(a\) to \(b\).
Understanding how to manipulate and integrate these expressions is a crucial part of solving calculus problems involving polar coordinates.

Definite Integrals

Definite integrals are used to calculate the area under a curve between two specified points, denoted by the integral limits. The definite integral of a function \(f(x)\) from \(a\) to \(b\) is represented as \(\int_{a}^{b} f(x) \, dx\).
In the context of polar coordinates, we often integrate with respect to \(\theta\), such as in the problem above, where we calculate the area \(A_n\) by integrating \(e^{-2\theta}\).
The reason for using definite integrals here is to calculate the specific area of a region bounded by consecutive turns of a spiral. The calculation involves determining the limits of \(\theta\) for these turns, solving the integral, and simplifying the expression.
Definite integrals are powerful tools not just for calculating areas but also for summing up quantities that accumulate at rates described by a function (such as velocity and distance).

Limits

Limits are a foundational concept in calculus that describe the behavior of a function as the input approaches a certain point. Understanding limits allows us to investigate the behavior of functions as they approach infinity or other critical points where they may not traditionally have values.
In the exercise, we evaluated \(\lim_{n \rightarrow \infty} A_n\), looking at how the area changes as you consider subsequent spirals indefinitely. Through the limit evaluation, we found that \(A_n\) approaches 0, signifying that the area shrinks with each additional spiral turn.
Conceptually, limits help us understand the behavior of a series or sequence over the long term. It's like zooming out to see the broader trend rather than individual data points.

Sequences and Series

Sequences and series are mathematical constructs that deal with lists of numbers, often defined by a specific rule, and the sums they form. A sequence is simply a collection of numbers ordered in a specific manner, while a series is the sum of the elements of a sequence.
In the exercise above, we are examining a sequence of areas \(A_n\) given by the polar integral. Evaluating the sequence involves understanding its behavior as \(n\) goes to infinity. The series related aspect comes into play when assessing the behavior (a recursive manner) of these areas and their ratios as exhibited by the limit \(\lim_{n \rightarrow \infty} A_{n+1}/A_n\).
This type of work with sequences and series is crucial for predicting and understanding how functions and numerical approximations behave, especially in calculus where these concepts often converge to produce meaningful results over infinite domains.