Problem 64 - Calculus Early Transcendentals

Question

Improper integrals arise in polar coordinates when the radial coordinate \(r\) becomes arbitrarily large. Under certain conditions, these integrals are treated in the usual way: $$\int_{\alpha}^{\beta} \int_{a}^{\infty} f(r, \theta) r d r d \theta=\lim _{b \rightarrow \infty} \int_{\alpha}^{\beta} \int_{a}^{b} f(r, \theta) r d r d \theta$$ Use this technique to evaluate the following integrals. $$\iint_{R} \frac{d A}{\left(x^{2}+y^{2}\right)^{5 / 2}} ; R=\\{(r, \theta): 1 \leq r < \infty, 0 \leq \theta \leq 2 \pi\\}$$

Step-by-Step Solution

Answer

Question: Evaluate the improper integral in polar coordinates of the function \(\frac{dA}{(x^2+y^2)^{5/2}}\) over the region where \(1 \leq r < \infty\) and \(0 \leq \theta \leq 2\pi\). Answer: The value of the improper integral is \(\frac{1}{4}\pi\).

Step 1: Identify the limits of integration in polar coordinates

The given region \(R\) is defined in polar coordinates as: $$1 \leq r < \infty, \qquad 0 \leq \theta \leq 2\pi$$ So, the radial coordinate will range from \(1\) to \(\infty\) and the angular coordinate will range from \(0\) to \(2\pi\).

Step 2: Convert the integral from Cartesian to polar coordinates

We need to rewrite the given integral in terms of polar coordinates. The area element in Cartesian is given by \(dA = dx dy\) which in terms of polar coordinates becomes: \(dA = r dr d\theta\). The integral function is given by \(\frac{dA}{(x^2+y^2)^{5/2}}\), so we need to replace \(x^2+y^2\) by \(r^2\). Therefore, the integral in polar coordinates becomes: $$\iint_{R} \frac{dA}{(r^2)^{5/2}} = \iint_{R} \frac{dA}{r^{10}}$$

Step 3: Set up the integral

Now we can set up the integral in polar coordinates using the region limits and function obtained in the previous steps: $$\int_{0}^{2\pi} \int_{1}^{\infty} \frac{r dr d\theta}{r^{10}}$$

Step 4: Apply the formula of improper integral

According to the provided formula for improper integrals in polar coordinates, we have: $$\int_{0}^{2\pi} \int_{1}^{\infty} \frac{r dr d\theta}{r^{10}} = \lim_{b \rightarrow \infty} \int_{0}^{2\pi} \int_{1}^{b} \frac{r dr d\theta}{r^{10}}$$

Step 5: Evaluate the integral

Now we can evaluate the integral as follows: $$\lim_{b \rightarrow \infty} \int_{0}^{2\pi} \int_{1}^{b} \frac{r dr d\theta}{r^{10}} = \lim_{b \rightarrow \infty} \int_{0}^{2\pi} \left[ -\frac{1}{8r^8} \right]_{1}^{b} d\theta = \lim_{b \rightarrow \infty} \int_{0}^{2\pi} \left(-\frac{1}{8b^8}+\frac{1}{8}\right) d\theta$$ $$= \lim_{b \rightarrow \infty} \left[\frac{1}{8}\theta\right]_{0}^{2\pi} \left(-\frac{1}{8b^8}+\frac{1}{8}\right) = \frac{1}{8}(2\pi - 0) \left(-\lim_{b \rightarrow \infty} \frac{1}{8b^8}+\frac{1}{8}\right) = \frac{1}{4}\pi.$$

Key Concepts

Polar Coordinates

Polar coordinates are a two-dimensional coordinate system where each point on a plane is determined by a distance from a reference point and an angle from a reference direction. In this system, the reference point is often called the pole, typically represented by the origin of a Cartesian plane, and the reference direction is usually the positive x-axis.

A point in polar coordinates is given by \( (r, \theta) \) where \( r \) is the radial distance from the pole, and \( \theta \) is the angular coordinate, i.e., the angle measured in radians from the reference direction. Compared to Cartesian coordinates \( (x, y) \) where the location is determined by x and y distances from the two perpendicular axes, polar coordinates offer a more natural approach to dealing with problems involving rotations, circles, and spirals.

Limits of Integration

In integral calculus, the limits of integration define the interval over which a function is to be integrated. When using polar coordinates, the radial limits are between \( r=a \) and \( r=b \) for some values of \( a \) and \( b \) and the angular limits are between \( \theta=\alpha \) and \( \theta=\beta \) for some angles \( \alpha \) and \( \beta \).

When dealing with areas where \( r \) extends infinitely, the upper limit for \( r \) is often written as \( \infty \) which signifies that the region extends outward indefinitely. To properly evaluate such integrals, techniques of improper integrals come into play, where we replace the infinite limit with a variable and take the limit of the resulting integral as the variable approaches infinity.

Converting Cartesian to Polar Coordinates

Conversion between Cartesian and polar coordinates is essential when dealing with problems that exhibit radial symmetry or are best described in terms of angles and distances from a point. To convert from Cartesian to polar coordinates, we use the following relations: \( r = \sqrt{x^2 + y^2} \) for the radial coordinate and \( \theta = \arctan(\frac{y}{x}) \) for the angular coordinate, where \( x \) and \( y \) are the Cartesian coordinates of the point.

For integrals, it's also important to note the area element change: the Cartesian area element \( dA = dx \, dy \) becomes \( r \, dr \, d\theta \) in polar coordinates to account for the varying distances of points from the pole—which increases the area they cover as \( r \) increases.

Evaluating Improper Integrals

Improper integrals occur when the function being integrated is unbounded or the interval of integration is infinite. In polar coordinates, this typically means that the radial coordinate \( r \) extends to \( \infty \) or that the function has a singularity within the region of integration. To evaluate these, we can't just plug in \( \infty \) — instead, we introduce a limit process.

We define the integral up to a certain finite value \( b \) and then take the limit as \( b \) approaches \( \infty \). This method allows us to calculate the value of an integral over an unbounded region, provided that the limit exists, which would signify that the integral converges to a specific value.

Integral Functions in Polar Form

When functions are represented in polar form, they align directly with the polar coordinate system making the evaluation of certain integrals more intuitive. For example, when integrating over a circular region or when dealing with functions that have radial symmetry, the polar form can dramatically simplify computations.

In the polar form, we express functions as \( f(r, \theta) \) and deal with the integration with respect to both \( r \) and \( \theta \). The presence of \( r \) in the differential area element \( r \, dr \, d\theta \) is crucial since it scales the area element properly thorough the radial distance from the pole, ensuring that the infinitesimal areas are calculated correctly throughout the integration process.