Problem 21 - Calculus for Biology and Medicine

Question

Determine whether each integral is convergent. If the integral is convergent, compute its value. $$ \int_{0}^{4} \frac{1}{x^{4}} d x $$

Step-by-Step Solution

Answer

The integral \( \int_{0}^{4} \frac{1}{x^4} \, dx \) is divergent due to the singularity at \( x = 0 \).

Step 1: Identify the Integral Bounds and Function

The integral given is \( \int_{0}^{4} \frac{1}{x^4} \, dx \). The bounds are from 0 to 4, and the function inside the integral is \( f(x) = \frac{1}{x^4} \). Note that the function \( \frac{1}{x^4} \) is undefined at \( x = 0 \), which may indicate a singularity or improper integral.

Step 2: Determine Convergence at the Lower Bound

Since the function has a singularity at \( x = 0 \), we need to analyze its behavior as \( x \) approaches zero. Rewriting the integral to consider the limit: \( \int_{0+}^{4} \frac{1}{x^4} \, dx \) becomes \( \lim_{a \to 0^+} \int_{a}^{4} \frac{1}{x^4} \, dx \). We will now integrate the function and then evaluate the limit.

Step 3: Integrate the Function

To integrate \( \frac{1}{x^4} \), we use the power rule for integration: \( \int \frac{1}{x^n} \, dx = \frac{x^{-n + 1}}{-n + 1} + C \). Thus, the antiderivative is \( -\frac{1}{3x^3} \). The definite integral becomes \[ \lim_{a \to 0^+} \left[ -\frac{1}{3x^3} \right]_{a}^{4}. \]

Step 4: Evaluate the Antiderivative at the Bounds

Substitute the bounds into the antiderivative: \[ -\frac{1}{3 \times 4^3} - \left( -\frac{1}{3a^3} \right) = -\frac{1}{192} + \frac{1}{3a^3}. \]

Step 5: Evaluate the Limit for Convergence

Now, evaluate the limit as \( a \to 0^+ \): \[ \lim_{a \to 0^+} \left( -\frac{1}{192} + \frac{1}{3a^3} \right). \] As \( a \to 0^+ \), \( \frac{1}{3a^3} \to \infty \). This implies that the integral diverges, as \( \lim_{a \to 0^+} \) yields infinity.

Key Concepts

Convergence and Divergence

When dealing with improper integrals, it's important to understand whether they converge or diverge. Convergence implies that the integral has a finite value, while divergence means it doesn't. The integral in the exercise, \( \int_{0}^{4} \frac{1}{x^4} \, dx \), exhibits a singularity at \( x = 0 \). Therefore, we need to check its convergence by taking the limit as \( x \) approaches zero from the right. Here,

• If the limit results in a finite value, the integral is convergent.
• If the limit is infinite or undefined, the integral is divergent.

In this case, the solution demonstrated that the integral diverges because as \( a \to 0^+ \), the term \( \frac{1}{3a^3} \) tends to infinity, leading the integral to increase indefinitely.

Singularities in Integrals

A singularity in an integral happens when the function within the integral becomes undefined or infinite at a particular point. In the integral \( \int_{0}^{4} \frac{1}{x^4} \, dx \), the singularity is at \( x = 0 \) since \( \frac{1}{x^4} \) approaches infinity when \( x \) is zero. This makes the integral improper. To manage this, we replace the singular portion with a limit approach:

• Rewriting the bound as a limit, \( \int_{0+}^{4} \frac{1}{x^4} \, dx \) becomes \( \lim_{a \to 0^+} \int_{a}^{4} \frac{1}{x^4} \, dx \).

This substitution lets us evaluate the behavior of the function as it nears the singular point, rather than including the point outright in the range of integration. By understanding where these singularities occur, we can better assess whether an integral will converge or diverge.

Power Rule for Integration

The power rule for integration is a fundamental tool used to find antiderivatives and evaluate integrals of power functions. The formula is:

• For functions in the form \( x^n \), the antiderivative is \( \frac{x^{n+1}}{n+1} + C \), where \( C \) is a constant of integration.
• For functions in the form \( \frac{1}{x^n} \), rewrite it as \( x^{-n} \) and apply the rule: \( \int x^{-n} \, dx = \frac{x^{-n+1}}{-n+1} + C \).

In our example, \( \int \frac{1}{x^4} \, dx \) was integrated by using the transformed function \( x^{-4} \), resulting in an antiderivative of \( -\frac{1}{3x^3} \). The power rule helped to find this by increasing the power index by one and dividing by the new index. Being comfortable with this rule is essential for solving many problems involving polynomial and power functions.