Problem 16 - Calculus and its applications

Question

Determine whether each improper integral is convergent or divergent, and find its value if it is convergent. $$ \int_{5}^{\infty} x^{4} d x $$

Step-by-Step Solution

Answer

The integral is divergent.

Step 1: Identify the Type of Improper Integral

The given integral \( \int_{5}^{\infty} x^{4} \, dx \) is an improper integral because it has an infinite upper limit. We need to evaluate the convergence by evaluating the limit of a definite integral replacing \( \infty \) with a finite limit \( b \). Then take \( b \to \infty \).

Step 2: Set Up the Integral with Finite Upper Bound

Replace \( \infty \) in the integral with a finite upper limit \( b \). Now the integral is \( \int_{5}^{b} x^{4} \, dx \). We will evaluate this integral first and then take the limit as \( b \to \infty \).

Step 3: Evaluate the Indefinite Integral

Integrate the function \( x^4 \) with respect to \( x \):\[ \int x^4 \, dx = \frac{x^5}{5} + C \] where \( C \) is the constant of integration.

Step 4: Apply the Definite Integral from 5 to b

Use the result from the indefinite integral to compute the definite integral:\[ \int_{5}^{b} x^4 \, dx = \left[ \frac{x^5}{5} \right]_{5}^{b} = \frac{b^5}{5} - \frac{5^5}{5} \] This simplifies to \( \frac{b^5}{5} - 125 \).

Step 5: Evaluate the Limit as b Approaches Infinity

Now, take the limit as \( b \to \infty \):\[ \lim_{b \to \infty} \left( \frac{b^5}{5} - 125 \right) = \lim_{b \to \infty} \frac{b^5}{5} - 125 \]As \( b \to \infty \), \( \frac{b^5}{5} \to \infty \), so the entire expression approaches \( \infty \).

Step 6: Conclusion: Determine Convergence or Divergence

Since the limit of the integral as \( b \to \infty \) is \( \infty \), the improper integral \( \int_{5}^{\infty} x^4 \, dx \) is divergent.

Key Concepts

Convergence and Divergence

When tackling improper integrals, determining whether they converge or diverge is crucial. This boils down to understanding if the integral results in a finite number (convergence) or grows without bound (divergence). For example, in the problem involving the integral from 5 to infinity of \(x^4\), we initially replace the infinite boundary with a finite number \(b\). Upon solving the integral and evaluating it as \(b\) approaches infinity, we observed that \(\frac{b^5}{5} - 125\) tends towards infinity. Therefore, the final step confirms divergence since the result isn't a fixed number. Remember:

• Convergence: Result is finite.
• Divergence: Result stretches infinitely.

Understanding this difference helps solve problems with infinite boundaries effectively.

Infinite Limits

The concept of infinite limits plays a crucial role when dealing with improper integrals. They help evaluate the behavior of functions as they extend towards infinity. In the exercise, we use infinite limits to replace \(\infty\) with \(b\) in the integral, producing \(\int_{5}^{b} x^4 \, dx\). This substitution helps analyze the function behavior in more manageable terms. By calculating the definite integral and letting \(b\) tend towards infinity, we find that the integral doesn’t settle at any finite value. This approach is essential for deciding if an integral converges or diverges. Consider infinite limits as a technique to "zoom out" and appreciate the broad landscape of a function's behavior. They allow substitution in initially intimidating problems with ease.

Definite Integrals

Definite integrals are fundamental in understanding improper integrals. They calculate the area under a curve over a specific interval. In the case of infinity-bound integrals, the definite integral from 5 to \(b\) involves calculating: \[\left[ \frac{x^5}{5} \right]_{5}^{b} = \frac{b^5}{5} - 125\]This process highlights how the limits of integration affect the result. It allows us to substitute upper bounds and assess the overall finite or infinite result. Remember, definite integrals offer the tools to quantify curves inclusively within any two boundaries, even when one is infinite. They anchor the floating concept of indefinite integration by providing a tangible result over a specified range. Always focus on how these apply within the bounds of your specific problem.