Problem 55 - Calculus: Early Transcendentals

Question

What is wrong with the equation? \( \displaystyle \int^1_{-2} x^{-4} \, dx = \frac{x^{-3}}{-3} \Bigg]^1_{-2} = -\frac{3}{8} \)

Step-by-Step Solution

Answer

Incorrectly evaluated; the integral diverges due to impropriety.

Step 1: Identify the Function and Limits

The given integral is \( \int_{-2}^{1} x^{-4} \, dx \). This is improper because the integrand \( x^{-4} \) becomes undefined at \( x = 0 \), which lies within the integration range \([-2, 1]\).

Step 2: Recognize the Impropriety

Since the integrand \( x^{-4} \) is undefined at \( x = 0 \), we must handle the integral as improper. It should be split into two parts and approached as a limit:\[ \int_{-2}^{1} x^{-4} \, dx = \lim_{c \to 0^-} \int_{-2}^{c} x^{-4} \, dx + \lim_{d \to 0^+} \int_{d}^{1} x^{-4} \, dx. \]

Step 3: Evaluate the Right Limit

Consider \( \lim_{d \to 0^+} \int_{d}^{1} x^{-4} \, dx \):The antiderivative is \( \frac{x^{-3}}{-3} \), so we write:\[ \lim_{d \to 0^+} \left( \frac{x^{-3}}{-3} \right) \Bigg|_{d}^{1} = \lim_{d \to 0^+} \left( \left( -\frac{1}{3} \right) - \left( -\frac{1}{3d^3} \right) \right). \]As \( d \) approaches \( 0 \), \( -\frac{1}{3d^3} \) tends to infinity, thus making the integral divergent.

Step 4: Conclusion on Improper Integral

The divergence from previous steps implies that the original integral \( \int_{-2}^{1} x^{-4} \, dx \) is divergent because one of the limits leads to an infinite value. Therefore, the calculation resulting in \(-\frac{3}{8}\) is incorrect.

Key Concepts

Divergent Integrals

Divergent integrals can be tricky because they involve values that approach infinity. One of the main problems with divergent integrals is that they occur when the function being integrated, or its limits, contains a point where the function goes infinite.In our case, the integral \( \int_{-2}^{1} x^{-4} \, dx \) is divergent because the integrand \( x^{-4} \) becomes infinite when \( x = 0 \), which is within the integration limits \([-2, 1]\).
To handle such situations, we must split the integral into parts and use limit processes to understand behavior near undefined points. This means calculating the integral from \(-2\) to a point approaching \(0\) from the left, and from another point approaching \(0\) from the right to \(1\).By using limits:

• Evaluate \( \lim_{c \to 0^-} \int_{-2}^{c} x^{-4} \, dx \)
• Evaluate \( \lim_{d \to 0^+} \int_{d}^{1} x^{-4} \, dx \)

These limits assess how the area under the curve behaves as you approach the points where the function becomes undefined.If either limit results in infinity, the integral is divergent.

Integration Techniques

Integration techniques are essential for solving complex integrals and ensuring accurate results.One common technique involves finding the antiderivative of the function: the reversed process of differentiation. In the problem, the integrand \( x^{-4} \) was integrated to obtain its antiderivative \( \frac{x^{-3}}{-3} \).
However, for improper integrals like this one, additional steps are required. Recognizing points where the function is undefined tells us how to apply limits.It’s important to understand when to use a direct approach and when to rely on limits due to undefined behavior. Here are some tips:

• If a function has undefined points in its integration range, consider breaking the integral into separate parts.
• Apply limits to classify the behavior as it approaches undefined points. This ensures precision.
• Double-check calculations; small errors in limits can lead to large errors in final values.

Being methodical about which technique to use and where can help prevent mistakes, especially in problems with more challenging integrals.

Calculus Errors

Errors in calculus, particularly with integration, can often arise from overlooking important details.In this exercise, the error was assuming that the integral \( \int_{-2}^{1} x^{-4} \, dx \) could be solved directly without considering its improper nature.
Such errors may arise if:

• The integrand is not checked for points where it could be undefined.
• Appropriate limit processes are ignored for improper integrals.
• Intermediate steps are not carefully verified.

In this problem, without using limit processes, one might wrongly conclude that the integral evaluates to \(-\frac{3}{8}\). But since one of the limits leads to infinity, the integral diverges, meaning it doesn’t converge to a specific number.
To prevent these errors, always verify:

• The domain and important behaviors of the function within its integration range.
• Applying correct limit calculations where necessary.
• Include checks for undefined behavior to ensure full integration accuracy.

Make these practices a habit to enhance your calculation accuracy and understanding of integral behavior.