Problem 20 - Calculus for Biology and Medicine

Question

Determine whether each integral is convergent. If the integral is convergent, compute its value. $$ \int_{1}^{\infty} \frac{1}{x^{1 / 3}} d x $$

Step-by-Step Solution

Answer

The integral diverges as it tends towards infinity.

Step 1: Identify the Type of Integral

The given integral is \( \int_{1}^{\infty} \frac{1}{x^{1 / 3}} \, dx \). This is an improper integral because it involves an infinite interval of integration. To determine if it converges, we need to analyze the behavior of the function \( \frac{1}{x^{1/3}} \) as \( x \to \infty \).

Step 2: Set Up the Limit Expression

To solve for convergence, express the integral as a limit: \[ \lim_{b \to \infty} \int_{1}^{b} \frac{1}{x^{1 / 3}} \, dx. \] We'll evaluate what happens as \( b \to \infty \).

Step 3: Integrate the Function

Find the integral of \( \frac{1}{x^{1/3}} \):\[ \int \frac{1}{x^{1/3}} \, dx = \int x^{-1/3} \, dx = \frac{x^{2/3}}{2/3} + C = \frac{3}{2} x^{2/3} + C. \]

Step 4: Apply the Limits of Integration

Apply the limits to the integrated function:\[ \lim_{b \to \infty} \left[ \frac{3}{2} x^{2/3} \right]_{1}^{b} = \lim_{b \to \infty} \left( \frac{3}{2}b^{2/3} - \frac{3}{2}(1)^{2/3} \right). \] Evaluate the expression as \( b \to \infty \).

Step 5: Evaluate the Convergence

Evaluate the limit:\[ \lim_{b \to \infty} \left( \frac{3}{2}b^{2/3} \right) - \frac{3}{2} = \infty - \frac{3}{2}. \] Since the expression tends towards infinity, the integral diverges.

Key Concepts

Convergence and Divergence

Improper integrals are integrals that deal with infinite limits or unbounded functions. One of the primary concerns when working with them is determining whether they converge or diverge. Convergence indicates that the integral has a finite value, while divergence implies that the value is infinite or undefined.
To understand convergence and divergence, we assess the behavior of the integral over its interval.

• If the function's integrated value tends to a finite number as the limits approach infinity, it converges.
• If not, it diverges, meaning the integral does not settle at a specific value.

This is crucial for integrals like \[\int_{1}^{\infty} \frac{1}{x^{1/3}} \ dx\]where the interval is infinite. We initially suspect divergence when the individual terms grow too large (like in this exercise). Understanding different types of behaviors helps identify if an integral will be convergent or divergent, informing mathematicians and students about the nature of the function they are dealing with.

Limit Evaluation

Evaluating limits is a fundamental process for determining an improper integral's behavior as it approaches infinity. When you set up a limit for an improper integral, such as:\[\lim_{b \to \infty} \int_{a}^{b} f(x) \, dx\]it allows you to observe how the function behaves as its domain stretches infinitely. In the case of the given problem, the limit expression is:\[\lim_{b \to \infty} \int_{1}^{b} \frac{1}{x^{1 / 3}} \, dx.\]When evaluating this expression, replace the integral with a primitive function. This often involves simplifying using techniques like differentiation or substitution. Once this simplified form is substituted back into the limit,we can then apply the bounds and observe the behavior as \( b \to \infty \).
This process determines the final value or conclusion on convergence or divergence by highlighting the trend the function follows in the realms beyond finite limits.

Integration Techniques

Integration techniques are crucial to finding the integral of a function and determining its convergence. Here, the fundamental approach involves finding the antiderivative.
For the integral\[\int \frac{1}{x^{1/3}} \, dx,\]you rewrite the function as \( x^{-1/3} \) to simplify. Then, apply the power rule for integration: \[\int x^n \, dx = \frac{x^{n+1}}{n+1} + C.\]For \( \frac{1}{x^{1/3}} \), or \( x^{-1/3} \), this becomes:\[\frac{x^{2/3}}{2/3} + C = \frac{3}{2} x^{2/3} + C.\]This integral then helps us establish values for limits, further feeding into limit evaluation. Integration techniques, like changing the expression form, are essential for handling complex integrals and providing a route to determine behaviors at infinity. By practicing these techniques, we can find relevant antiderivatives, apply limits appropriately, and ultimately decide on convergence or divergence efficiently.