Problem 12 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Determine whether the integral is convergent or divergent. Evaluate all convergent integrals. Be efficient. If \(\lim _{x \rightarrow \infty} \neq 0\), then \(\int_{a}^{\infty} f(x) d x\) is divergent. \(\int_{0}^{\infty} x d x\)

Step-by-Step Solution

Answer

The integral \(\int_{0}^{\infty} x d x\) is divergent.

Step 1: Compute the Integral

The integral of \(x\) with respect to \(x\) is \(0.5x^2\). So, \(\int_{0}^{\infty} x d x\) becomes \(\lim_{{b} \to \infty} [0.5b^2 - 0]\).

Step 2: Evaluate the Limit

As \(b\) goes to infinity, \(0.5b^2\) goes to infinity. Therefore, \(\lim_{{b} \to \infty} [0.5b^2 - 0]\) equals infinity.

Step 3: Determine Convergence or Divergence

Since the limit as \(b\) goes to infinity equals infinity and does not result in a specific real number, the given integral \(\int_{0}^{\infty} x d x\) is divergent.

Key Concepts

Improper Integrals

Improper integrals are a special type of integral where we deal with unbounded intervals or unbounded functions. This means the limits of integration can extend to infinity, or the function itself might become infinite over the interval. These situations make it hard to evaluate the integral using normal methods. To solve improper integrals, we often use limits to "tame" these infinities. This allows us to see if the integral converges to a specific value or diverges to infinity. For example, evaluating \[\int_{0}^{\infty} x \, dx\] involves taking the limit as one of the bounds approaches infinity.Some points to note about improper integrals:

• If the limit results in a finite number, the integral is said to be convergent.
• If the limit heads towards infinity or does not exist, the integral is divergent.

Convergence or divergence defines whether we can assign a finite value to the integral or not. So, understanding improper integrals is crucial in contexts where we must interpret how infinite processes accumulate over a range.

Integral Calculus

Integral calculus is a branch of calculus focused on the concept of integration. It provides tools to compute integrals, which you can think of as the "area under a curve." The basic goal of integration is to find the total accumulated value, which can be areas, volumes, or other quantities depending on the application.Integration is the reverse process of differentiation, and there are many methods to compute integrals. For instance:

• Definite Integrals: Used to find the exact area under a curve from one point to another, like \(\int_{a}^{b} f(x)\, dx\).
• Indefinite Integrals: Represent families of functions and include an arbitrary constant, like \(\int f(x)\, dx + C\).

In integral calculus, the fundamental theorem of calculus connects differentiation with integration, showing that they are inverse operations. This connection is central to solving problems in calculus.

Limits at Infinity

Limits at infinity involve evaluating a function as the input grows very large or very small. When dealing with integrals extending to infinity, understanding limits at infinity is essential.Let’s break down what limits at infinity mean for an integral like \[\int_{0}^{\infty} x \, dx\].

• We replace the upper bound with a variable, typically \(b\), and consider the limit as \(b\) approaches infinity.
• We compute the integral normally, but in the final step, take the limit of our result as the variable approaches infinity, which helps us determine if the integral converges or diverges.

In the given problem, as the limit \[\lim_{{b} \to \infty} [0.5b^2 - 0]\]grows without bound, we conclude that the integral diverges. Grasping limits at infinity helps us assess the behavior of functions as they reach extreme values and plays a vital role in determining potential convergence or divergence of integrals.