Question

Determine whether the improper integrals converge or diverge. If possible, determine the value of the integrals that converge.\(\int_{0}^{2} \frac{d x}{x^{3}}\)

Short answer

The integral diverges because the limit results in infinity as we approach 0.

Step-by-step solution

Identify the Type of Improper Integral

The integral \( \int_{0}^{2} \frac{dx}{x^3} \) is an improper integral because the integrand \( \frac{1}{x^3} \) is not defined at \( x = 0 \). Thus, we need to evaluate it as a limit as \( x \) approaches 0 from the positive side.

Set Up the Limit

We express the integral as a limit: \(\lim_{t \to 0^+} \int_{t}^{2} \frac{1}{x^3} \, dx.\)

Evaluate the Integral

Calculate the indefinite integral of \( \frac{1}{x^3} \) which is \( -\frac{1}{2x^2} \). Therefore, the definite integral becomes: \(-\frac{1}{2x^2} \Big|_t^2.\)

Compute the Limit

Substitute the limits into the evaluated integral: \(\lim_{t \to 0^+} \left(-\frac{1}{2(2)^2} + \frac{1}{2t^2}\right).\)This simplifies to: \(\lim_{t \to 0^+} \left(-\frac{1}{8} + \frac{1}{2t^2}\right)\).As \( t \) approaches 0, \( \frac{1}{2t^2} \) approaches infinity, making the integral diverge.

Key concepts

Divergence and Convergence

When dealing with improper integrals, understanding the concepts of divergence and convergence is crucial. These terms describe the behavior of an integral as its limits approach certain values, notably infinity or points of discontinuity. An integral converges when it reaches a finite number as we evaluate its limit. Conversely, it diverges when the limit doesn't result in a finite value. For example, if plugging in values into the integrated function leads to an infinite result, then the integral diverges.
In the case of the integral \( \int_{0}^{2} \frac{dx}{x^3} \), the integral is improper because the function \( \frac{1}{x^3} \) is undefined at \( x=0 \). As we evaluate the limit approaching zero, the result tends towards infinity. This indicates that the integral diverges. Knowing whether an integral converges or diverges helps in predicting the sum of areas, particularly when the function doesn't behave well near the limits.

• Convergence: Finite limit
• Divergence: Infinite limit
• Improper integral: Needs evaluation over limit due to point of discontinuity or infinity

Limits in Calculus

Limits are fundamental in calculus, especially when handling improper integrals. They help us understand the behavior of functions as they approach particular values, often boundaries of integration. To compute an improper integral like \( \int_{0}^{2} \frac{dx}{x^3} \), limits allow us to bypass undefined regions by "creeping" up to those points without actually landing on them.
For instance, because \( x = 0 \) causes a division by zero in \( \frac{1}{x^3} \), we approach zero from the positive side using a limit. We represent this as \( \lim_{t \to 0^+} \int_{t}^{2} \frac{1}{x^3} \ dx \). The limit in calculus skillfully handles these undefined points, letting us understand the integral's behavior near points of discontinuity. Trying to grasp limits better can simplify the process of solving problems involving continuity and differentiability.

• Role of Limits: Evaluate undefined or infinite points
• Approach without reaching: Arrow notation \( t \to 0^+ \) signifies getting close
• Application: Helps compute improper integrals and their behavior

Evaluating Limits

Evaluating limits is pivotal when solving improper integrals. It involves calculating the tendency of a function as it approaches a specific value. Consider the process involved in \( \lim_{t \to 0^+} \left(-\frac{1}{2(2)^2} + \frac{1}{2t^2}\right) \). Here, evaluating means that as \( t \) gets very, very small, we assess where the expression heads.
The challenge often lies in determining whether the expression converges to a finite limit or diverges to infinity. In our example, because \( \frac{1}{2t^2} \) tends towards infinity as \( t \to 0^+ \), the integral overall diverges. Through evaluation, we confirm divergence in our calculation.

• Assessing Trends: Keep an eye on infinite or finite behavior
• Math Notation: Limits written to denote close approach, e.g., \( t \to 0^+ \)
• Outcome: Decide between convergence and divergence, impacting the understanding of integrals