Problem 40 - Calculus Early Transcendentals: Pearson New International Edition

Question

Suppose that \(f\) is continuous on \([0, \infty)\) except at \(x=1\), where \(\lim _{x \rightarrow 1}|f(x)|=\infty\). How would you define $\int_{0}^{\infty} f(x) d x ? \quad

Step-by-Step Solution

Answer

The integral is split at the discontinuity; evaluate as \( \lim_{a \to 1^-} \int_{0}^{a} f(x) \, dx + \lim_{b \to 1^+} \int_{b}^{\infty} f(x) \, dx \).

Step 1: Understanding the Context of the Integral

Given that the function \( f(x) \) is continuous on \([0, \infty)\) except at \( x=1 \), and \( \lim_{x \rightarrow 1}|f(x)|=\infty \), the function has an infinite discontinuity at \( x=1 \). We must deal with this discontinuity when defining the integral from 0 to infinity.

Step 2: Splitting the Integral into Parts

To handle the infinite discontinuity, we split the integral \( \int_{0}^{\infty} f(x) \, dx \) into two separate integrals: \( \int_{0}^{1} f(x) \, dx \) and \( \int_{1}^{\infty} f(x) \, dx \).

Step 3: Defining the Improper Integral Around the Discontinuity

Since \( f(x) \) is not continuous at \( x=1 \), we must consider limits to evaluate integrals around this point. Specifically, define \( I_1 = \int_{0}^{1} f(x) \, dx \) as \( \lim_{a \to 1^-} \int_{0}^{a} f(x) \, dx \) and \( I_2 = \int_{1}^{\infty} f(x) \, dx \) as \( \lim_{b \to 1^+} \int_{b}^{\infty} f(x) \, dx \).

Step 4: Combining the Integrals

To define the overall integral \( \int_{0}^{\infty} f(x) \, dx \), combine the two improper integrals as \( \lim_{a \to 1^-} \int_{0}^{a} f(x) \, dx + \lim_{b \to 1^+} \int_{b}^{\infty} f(x) \, dx \). Each integral must converge for their sum to be defined.

Key Concepts

Infinite Discontinuity

When dealing with integrals, an infinite discontinuity occurs at a point where a function approaches infinity, creating a vertical asymptote in the graph. For example, if we have a function like \( f(x) \) that is continuous on \([0, \infty)\), but at \(x=1\), \( \lim_{x \rightarrow 1}|f(x)|=\infty\), this means the function is not defined or goes towards infinity at this point.
Infinite discontinuities are crucial in calculus as they require us to apply special techniques to evaluate integrals. It complicates the process because standard integral calculation assumes continuity. Therefore, we must adjust our approach to properly evaluate any integrals that pass through or approach this point.
In practice, this often involves splitting the integral at the point of discontinuity, allowing us to handle each section where the function is better-behaved separately. Without addressing these points of infinite discontinuity, we wouldn't be able to correctly define or calculate the integral's value.

Limit Definition of Integrals

The limit definition of integrals becomes crucial when you have improper behavior in functions, such as infinite discontinuities. In these situations, a standard approach to integrate doesn't apply, so we utilize limits to ensure the integral is well-defined.
For the problematic section of the integral, such as near a discontinuity, we replace the point with a limit process. We define the integral near the discontinuity separately, such as \( \int_{0}^{1} f(x) \, dx = \lim_{a \to 1^-} \int_{0}^{a} f(x) \, dx \).

This means that instead of directly integrating up to the point \( x = 1 \), we approach it gradually from the left using \( a \) as a variable that gets very close to 1, but never reaches it. This same process applies to the other half from 1 to infinity as \( \int_{1}^{\infty} f(x) \, dx = \lim_{b \to 1^+} \int_{b}^{\infty} f(x) \, dx \).
Using limits allows us to manage how the function behaves as it approaches the points of infinite discontinuity. This approach effectively breaks the complex behavior into more manageable sections where each section can be independently evaluated.

Convergence of Integrals

Convergence is vital when evaluating improper integrals, particularly those involving infinite discontinuities. A convergent integral means that as we approach infinite limits or discontinuity points, the integral settles towards a finite value. If it doesn’t, it diverges, indicating that the integral does not have a defined value.
For a practical example, after splitting the integral to manage the infinite discontinuity, we must assess if each part converges. This means examining \( \lim_{a \to 1^-} \int_{0}^{a} f(x) \, dx \) and \( \lim_{b \to 1^+} \int_{b}^{\infty} f(x) \, dx \). Each of these limits must produce a finite result.
If either part does not converge to a finite number, the overall integral \( \int_{0}^{\infty} f(x) \, dx \) isn't defined. Therefore, the concept of convergence helps define whether an improper integral, affected by infinite discontinuities, is meaningful, allowing us to fully understand and utilize such integrals in mathematics.