Problem 38 - Thomas Calculus

Question

In polar coordinates, the average value of a function over a region \(R\) (Section 15.3\()\) is given by $$\frac{1}{\text { Area(R) }} \iint_{R} f(r, \theta) r d r d \theta$$ Converting to a polar integral Integrate \(f(x, y)=\) \(\left[\ln \left(x^{2}+y^{2}\right)\right] /\left(x^{2}+y^{2}\right)\) over the region \(1 \leq x^{2}+y^{2} \leq e^{2}\)

Step-by-Step Solution

Answer

The average value is \( \frac{2}{e^2 - 1} \).

Step 1: Identify Conversion to Polar Coordinates

The given function is in Cartesian coordinates: \( f(x, y) = \frac{\ln(x^2 + y^2)}{x^2 + y^2} \). In polar coordinates, the relation between Cartesian and polar coordinates is given by \( x = r \cos \theta \) and \( y = r \sin \theta \). Consequently, \( x^2 + y^2 = r^2 \). Therefore, the function converts to \( f(r, \theta) = \frac{\ln(r^2)}{r^2} \).

Step 2: Simplify Function in Polar Form

Using the identity \( \ln(r^2) = 2 \ln(r) \), transform the function in polar coordinates: \( f(r, \theta) = \frac{2 \ln(r)}{r^2} \).

Step 3: Define Integration Limits

In the given problem, the integration bounds for \( r \) are determined from the region \( 1 \leq x^2 + y^2 \leq e^2 \), which translates to \( 1 \leq r^2 \leq e^2 \) or \( 1 \leq r \leq e \). As the problem involves a full circle, \( \theta \) ranges from \( 0 \) to \( 2\pi \).

Step 4: Set Up the Polar Integral

The integral for the average value in polar coordinates is \( \frac{1}{\text{Area}(R)} \iint_R f(r, \theta) \, r \, dr \, d\theta \). First calculate the polar integral: \( \int_{0}^{2\pi} \int_{1}^{e} \left( \frac{2 \ln(r)}{r^2} \right) r \, dr \, d\theta \). Simplifying gives \( \int_{0}^{2\pi} \int_{1}^{e} \frac{2 \ln(r)}{r} \, dr \, d\theta \).

Step 5: Evaluate the Inner Integral

We first evaluate \( \int_{1}^{e} \frac{2 \ln(r)}{r} \, dr \). Let \( u = \ln(r) \), then \( du = \frac{1}{r} \, dr \). Thus, the integral becomes \( \int_{\ln(1)}^{\ln(e)} 2u \, du = \left. u^2 \right|_{0}^{1} = 1 \).

Step 6: Evaluate the Outer Integral

The outer integral is now \( \int_{0}^{2\pi} 1 \, d\theta \), which equals \( \theta \) evaluated over \( 0 \) to \( 2\pi \); thus, it results in \( 2\pi \).

Step 7: Compute the Area of Region R

The area of the annular region is given by the difference in the areas of the larger and smaller circles: \( \pi e^2 - \pi = \pi(e^2 - 1) \).

Step 8: Calculate the Average Value

Combine the results: \( \frac{1}{\pi (e^2 - 1)} \cdot 2\pi = \frac{2\pi}{\pi(e^2 - 1)} = \frac{2}{e^2 - 1} \).

Key Concepts

Polar Coordinates

Polar coordinates are a system where each point in a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point, typically called the origin, is where the distance, denoted as \( r \), is measured from. The angle, \( \theta \), is measured from a fixed direction, commonly the positive x-axis.

This system is particularly useful for regions that have circular symmetry, as it simplifies the equations that describe circles and spheres. For example, every point on a circle with center at the origin can be represented by the same angle but varying \( r \).

In converting from Cartesian (x, y) coordinates to polar (r, \( \theta \)) coordinates, the transformations \( x = r \cos \theta \) and \( y = r \sin \theta \) are used. Additionally, \( x^2 + y^2 = r^2 \) is a key relationship that simplifies many integrals, as seen in our exercise.

Average Value of a Function

The average value of a function across a specific region helps find a single value that represents the function's overall behavior over that area. In mathematical terms, it involves integrating the function over the region and dividing by the size of the region to "normalize" the value.

This can be directly connected to the formula for average value in polar coordinates: \[ \frac{1}{\text{Area}(R)} \iint_R f(r, \theta) \, r \, dr \, d\theta. \]

Here, \( f(r, \theta) \) is the function we are averaging, and \( R \) is the region over which we integrate. The average value formula involves a double integral, which signifies that we're considering both dimensions of the area. This gives a true representation of the average when examining multi-dimensional spaces.

Such calculations are often used in physics and engineering to find averagely distributed forces or quantities.

Change of Variables

The change of variables is a mathematical technique used to simplify complex integrals. This method involves substituting new variables that make the problem easier to solve. Changing from Cartesian coordinates to polar coordinates is a prime example.

When changing variables from \( x, y \) to \( r, \theta \), we account for the area distortion by substituting \( r \) instead of straight up using \( dx \) and \( dy \). This is because when you convert to polar, each \( dr \, d\theta \) rectangle transforms into a sector-like area with radius \( r \), thus incorporating an \( r \) in the integrand.

This technique is not only about simplifying calculations, it can also provide insights into the behavior of the function over different areas. This is crucial when dealing with symmetries or irregular regions that align better with polar coordinates.

Double Integrals

Double integrals are used to calculate the accumulation of values over a two-dimensional area. They extend the concept of a definite integral, which accumulates values over an interval along a single dimension.

In terms of polar coordinates, a double integral is expressed as: \[ \int_{a}^{b} \int_{c}^{d} f(r, \theta) \, r \, dr \, d\theta. \]

These integrals consider both the radial and angular changes, effectively covering a region in the plane. This process involves evaluating the inner integral, which goes along the radius \( r \), and subsequently the outer integral, which runs along the angle \( \theta \).

Double integrals frequently find applications in physical sciences and engineering, for calculating masses, areas, and the center of mass. Understanding how to perform these integrals, especially in polar coordinates, is a fundamental skill for analyzing and interpreting spatial data.