Problem 68 - Applied Calculus

Question

Does \(\int_{0}^{\infty} \frac{1}{1000} d x\) converge or diverge?

Step-by-Step Solution

Answer

The integral diverges.

Step 1: Understanding the Problem

We need to determine whether the improper integral \( \int_{0}^{\infty} \frac{1}{1000} dx \) converges or diverges. An integral over an infinite interval like this one can either accumulate to a finite value (converge) or become infinite (diverge).

Step 2: Visualize the Function

The function given in the integral is \( f(x) = \frac{1}{1000} \), which is a constant function. It plots as a horizontal line at \( y = \frac{1}{1000} \) for all \( x \geq 0 \).

Step 3: Setting Up the Definition

Since the integral is improper due to its infinite upper limit, we express it using a limit: \[ \int_{0}^{\infty} \frac{1}{1000} dx = \lim_{b \to \infty} \int_{0}^{b} \frac{1}{1000} dx \]

Step 4: Evaluate the Integral

Calculate the definite integral \( \int_{0}^{b} \frac{1}{1000} dx \). The antiderivative of \( \frac{1}{1000} \) with respect to \( x \) is \( \frac{x}{1000} \). Therefore, the integral evaluates to \[ \int_{0}^{b} \frac{1}{1000} dx = \left[ \frac{x}{1000} \right]_{0}^{b} = \frac{b}{1000} - \frac{0}{1000} = \frac{b}{1000} \]

Step 5: Take the Limit

Now substitute back into the limit expression: \[ \lim_{b \to \infty} \frac{b}{1000} \]. As \( b \to \infty \), the result \( \frac{b}{1000} \) clearly goes to infinity. Thus, the integral diverges.

Key Concepts

Convergence and Divergence

When dealing with improper integrals, one key question is whether they "converge" or "diverge". This is a way to determine if the integral results in a finite number or shoots off to infinity. For an integral to converge, the area under the curve must accumulate to a finite value.

• Convergence occurs if the value of the integral approaches a fixed number as the upper limit goes to infinity.
• Divergence occurs if the integral grows without bound, approaching infinity instead of settling on a finite value.

In the original exercise, despite the function being a constant and seemingly tame, the area under the infinite span clearly becomes enormous because the integral results in infinity. Hence it diverges.

Constant Function

A constant function is one that remains the same regardless of the input's value. In our case, the given function is \(f(x) = \frac{1}{1000}\). This function is simple because it neither rises nor falls. It makes a horizontal line on a graph.

• Constant functions are essentially flat; for all inputs \(x\), the output is the exact same value.
• Even though in this case the function \( \frac{1}{1000} \) seems small, over an infinite interval, it accumulates unlimited area beneath it.

Because the function is constant, evaluating its integral over a finite section is easy. However, infinite sections lead to divergence due to the nature of constant area accumulation.

Limit of Integration

In integrals that stretch to infinity, we use the concept of a limit to handle what we cannot calculate directly. The limit allows us to approach infinity in a mathematical sense, instead of trying to "reach" it.

• We express the improper integral \( \int_{0}^{\infty} \) as \( \, \lim_{b \to \infty} \int_{0}^{b} \).
• This method of using a limit partitions the problem into manageable sections and reflects how the integral would behave as the interval grows indefinitely.

In practice, we solve the integral over a finite section and then take the limit. In this exercise, by evaluating \( \int_{0}^{b} \frac{1}{1000} \, dx \) and then taking the limit, we see that as \( b \) increases indefinitely, the integral result increases without bounds. This use of limits shows that, despite being a constant function, the area continues to grow, leading to divergence.