Question

Consider an infinitely long wire coinciding with the positive \(x\) -axis and having mass density \(\delta(x)=\left(1+x^{2}\right)^{-1}\), \(0 \leq x<\infty\) (a) Calculate the total mass of the wire. (b) Show that this wire does not have a center of mass.

Short answer

(a) The total mass is \( \frac{\pi}{2} \). (b) The wire does not have a center of mass because the integral diverges.

Step-by-step solution

Define the Integral for Total Mass

The total mass of the wire can be found by integrating the mass density function over the range from 0 to infinity. Therefore, we set up the integral as \( \int_{0}^{\infty} \frac{1}{1+x^{2}}\, dx \).

Recognize the Formula

The integral \( \int \frac{1}{1+x^2} \, dx \) is a well-known trigonometric integral that simplifies to \( \arctan(x)+C \).

Evaluate the Integral

Now, we evaluate the indefinite integral and apply the limits from 0 to infinity: \[ \lim_{b \to \infty} \left( \arctan(b) - \arctan(0) \right). \] Since \( \arctan(\infty) = \frac{\pi}{2} \) and \( \arctan(0) = 0 \), the total mass is \( \frac{\pi}{2} \).

Set up the Integral for Center of Mass

The center of mass along the x-axis can be found using the formula \( x_{cm} = \frac{1}{M} \int_{0}^{\infty} x \delta(x) \, dx \), where \( M \) is the total mass, which is \( \frac{\pi}{2} \). Substitute \( \delta(x) \) to get \( x_{cm} = \frac{2}{\pi} \int_{0}^{\infty} \frac{x}{1+x^2} \, dx \).

Simplification of the Integral for x_cm

Recognize that \( \int \frac{x}{1+x^2} \, dx = \frac{1}{2} \ln(1+x^2)+C \). To evaluate the integral from 0 to \( \infty \), calculate \( \lim_{b \to \infty} \left( \frac{1}{2} \ln(1+b^2) - \ln(1+0^2) \right). \) As \( b \to \infty\), \( \ln(1+b^2) \to \infty\), making the integral diverge.

Conclusion about the Center of Mass

Since \( \int_{0}^{\infty} \frac{x}{1+x^2} \, dx \) diverges, the center of mass \( x_{cm} \) does not exist. The wire does not have a finite center of mass due to the divergence of this integral.

Key concepts

Integral Calculus

Integral calculus is a branch of calculus that is centered around the concept of integration. Integration is used to calculate areas, volumes, central points, and many useful things. In this exercise, integral calculus helps us determine the total mass of the wire by integrating the mass density function over a specified range.
Integrating a function involves finding the integral, an expression that gives the area under a curve defined by that function. When calculating the total mass of the wire, we set up the integral of the mass density function, \(\delta(x) = \frac{1}{1+x^{2}}\), over the range from 0 to infinity.
This process not only provides the total mass but also reveals the importance of integral bounds and convergence in achieving accurate results. Proper evaluation ensures that every part of the infinite region is accounted for effectively.

Mass Density

Mass density, denoted as \(\delta(x)\), is a measure of the mass per unit length along the wire. It is crucial for calculating the total mass and understanding the distribution of mass in the wire. In this problem, the mass density is given by the function \(\delta(x) = (1 + x^2)^{-1}\), which decreases as you move further along the wire.
This specific function implies that more mass is concentrated near the start of the wire, and it gets progressively lighter with distance. The task involves integrating this function over the entire length of the wire to find its total mass.
Understanding mass density helps in setting up integrals properly, ensuring that we incorporate every unit of mass along the infinite extension of the wire effectively.

Trigonometric Integrals

Trigonometric integrals are a category of integrals involving trigonometric functions. They often require specific techniques or knowledge of formulas, as seen in this exercise, where the integral of the form \(\int \frac{1}{1+x^2} \, dx \) simplifies to \(\arctan(x) + C\).
The presence of trigonometric terms indicates the necessity to use inverse trigonometric functions to evaluate these integrals. In this particular problem, recognizing the integral as a standard form allows us to apply the inverse tangent function, \(\arctan(x)\), to compute the total mass.
Mastering trigonometric integrals involves recognizing patterns and recalling formulas, which can considerably simplify the process of solving complex integral calculus problems.

Improper Integrals

Improper integrals involve limits that extend to infinity or include discontinuities, which is the case for our current evaluation. They occur when the range of integration extends infinitely, or the function being integrated approaches infinity within the limits.
In this exercise, we have two instances of improper integrals. The first is calculating the total mass over an infinite domain, and the second involves determining whether the center of mass exists. In both cases, the integral is evaluated as a limit: \(\lim_{b \to \infty} \), addressing the improper nature.
For the center of mass, the improper integral diverges, implying no finite solution exists. Understanding improper integrals aids in tackling advanced calculus problems, particularly those involving infinite limits in real-world scenarios.