Problem 24 - Calculus Single Variable

Question

Determine whether the given improper integral is convergent or divergent. If it converges, then evaluate it. \(\int_{3}^{6}(x-4)^{-2} d x\)

Step-by-Step Solution

Answer

The integral is divergent.

Step 1: Identify the Integral Type

The given integral is \( \int_{3}^{6} (x-4)^{-2} \, dx \). Since the function \((x-4)^{-2}\) becomes undefined at \(x = 4\) (within the integration bounds), this integral is improper due to the discontinuity within the interval. We need to check for convergence around this point.

Step 2: Set Up Limits for the Discontinuity

To handle the discontinuity at \(x = 4\), we split the integral into two parts and use limits:\[\int_{3}^{6} (x-4)^{-2} \, dx = \int_{3}^{4} (x-4)^{-2} \, dx + \int_{4}^{6} (x-4)^{-2} \, dx\]We then evaluate each part with a limit as \(x\) approaches 4.

Step 3: Evaluate the Left-Side Integral

For \(\int_{3}^{4} (x-4)^{-2} \, dx\), substitute \( t = x - 4 \), then \( dx = dt \) and limits change to \( t = -1 \) to \( t = 0 \). The integral becomes (using limits):\[\lim_{b \to 4^{-}} \int_{-1}^{b-4} t^{-2} \, dt = \lim_{b \to 4^{-}} \left[ -t^{-1} \right]_{-1}^{b-4}.\]Evaluate the limit as \(b\) approaches 4 from the left.

Step 4: Solve the Left-Side Limit

The limit becomes:\[\lim_{b \to 4^{-}} \left( -\frac{1}{b-4} + 1 \right).\]As \(b\) approaches 4 from the left, \(b-4\) approaches 0 from the negative side, and \(-\frac{1}{b-4}\) becomes unbounded (tends to positive infinity). Hence, \(\int_{3}^{4} (x-4)^{-2} \, dx\) diverges.

Step 5: Analyze the Right-Side Integral

Although the left-side integral diverges, for completeness:For \(\int_{4}^{6} (x-4)^{-2} \, dx\), substitute \( t = x - 4 \), then \( dx = dt \) and bounds change to \( t = 0 \) to \( t = 2 \). This becomes:\[\lim_{a \to 4^{+}} \int_{a-4}^{2} t^{-2} \, dt = \lim_{a \to 4^{+}} \left[- t^{-1} \right]_{a-4}^{2}\],which evaluates to a convergent value, but is overshadowed by the left divergence.

Step 6: Conclusion

Since we found that one part of the integral diverges, the entire integral \( \int_{3}^{6} (x-4)^{-2} \, dx \) is divergent regardless of the other part.

Key Concepts

Convergence

Convergence in the context of integrals refers to the condition under which an improper integral sums up to a finite, specific value. For proper integrals, convergence means the limit of the integral sums to a specific number. However, with improper integrals, where the function might have discontinuities or infinite intervals, special care is required to determine convergence.

When dealing with improper integrals, you must carefully examine points of discontinuity in the function and evaluate the behavior around them. This often requires splitting the integral into manageable parts and using limits to approach problematic points.

For example, in the exercise given, the integral \[\int_{3}^{6} (x-4)^{-2} \, dx\]is improper because \((x-4)^{-2}\) becomes infinite at \(x = 4\). By evaluating the limits as the integration variable approaches problematic points, you can check if the integral converges to a finite value.

Divergence

Divergence occurs when an integral does not yield a finite number. Instead, as you evaluate the integral around points of discontinuity or infinity, the result tends to infinity or fails to settle at a definite number.

In the exercise under review, we note that around \(x = 4\), the expression \((x-4)^{-2}\) exhibits a behavior that prevents it from converging. When evaluated, one side of the split integral resulted in a term becoming infinitely large as it approached the discontinuity at zero, leading to divergence.

This is a key point: although one part of an improper integral may converge separately, if any part diverges, the entire integral is classified as divergent. Recognizing divergence is crucial for understanding the overall behavior of the integral and ensuring proper mathematical conclusions.

Limits

Limits are a foundational tool in calculus, used to evaluate the behavior of functions at a point, especially where functions are undefined. They help in handling points of discontinuity by approaching these points arbitrarily closely from either the left or the right.

The concept of limits was used strategically in the given integral problem. To address the point of discontinuity at \(x = 4\), the integral was split and each part was evaluated using limits. For the left-sided integral from \(3\) to \(4\), a limit was set up:

• As \(b \to 4^{-},\) the integral \(\int_{3}^{b} (x-4)^{-2}\, dx\) was examined by replacing \(x\) with \(t = x - 4\).
• Similarly, for the right-sided integral from \(4\) to \(6\), limits were applied as \(x\) approached \(4\) from the right.

By applying such limits, one can more accurately determine the behavior of an integral near troublesome points. This makes limits an invaluable method in analyzing improper integrals.

Discontinuous Functions

Discontinuities present a unique challenge in mathematics, particularly when evaluating integrals. A function is discontinuous at a point if it is not well-defined or jumps at that location. In the case of integrals, these discontinuities need special handling, usually involving limits, to properly assess the area under the curve.

In our case, the function \((x-4)^{-2}\) becomes undefined precisely at \(x = 4\). This creates what is known as a vertical asymptote, where the function spikes infinitely. The presence of this point within the bounds of integration converts the given integral into an improper one.

To manage this, the integral was broken into two segments that approached continuity: one approaching from below \(x = 4\) and the other above. Each segment was then analyzed individually using limits to assess convergence or divergence. Understanding how to deal with such discontinuities is crucial to effectively working with and solving improper integrals.