Problem 17 - Calculus for Biology and Medicine

Question

In Problems \(17-28\), determine whether each integral is convergent. If the integral is convergent, compute its value. $$ \int_{1}^{\infty} \frac{1}{x^{3}} d x $$

Step-by-Step Solution

Answer

The integral is convergent and its value is \( \frac{1}{2} \).

Step 1: Identify the Type of Integral

The integral given is \( \int_{1}^{\infty} \frac{1}{x^3} \, dx \). This is an improper integral because it has an infinite upper limit. We will need to find whether it is convergent by evaluating the limit of a definite integral as it approaches infinity.

Step 2: Set Up the Limit for the Improper Integral

Convert the improper integral into a proper one with the limit notation:\[\lim_{b \to \infty} \int_{1}^{b} \frac{1}{x^3} \, dx\]

Step 3: Integrate the Function

Find the antiderivative of \( \frac{1}{x^3} \). The antiderivative is \(-\frac{1}{2x^2}\). Integrating \( \frac{1}{x^3} \) from 1 to \( b \) gives:\[\left[-\frac{1}{2x^2}\right]_{1}^{b} = -\frac{1}{2b^2} + \frac{1}{2}\]

Step 4: Evaluate the Limit

Evaluate the limit as \( b \to \infty \):\[\lim_{b \to \infty} \left(-\frac{1}{2b^2} + \frac{1}{2}\right)\]Here, \(-\frac{1}{2b^2}\) approaches 0 as \( b \to \infty \), thus the expression simplifies to \( \frac{1}{2} \).

Step 5: Conclude the Convergence

Since the limit exists and is a finite number (\( \frac{1}{2} \)), the integral \( \int_{1}^{\infty} \frac{1}{x^3} \, dx \) is convergent and its value is \( \frac{1}{2} \).

Key Concepts

Improper Integral

An improper integral is where the function we're integrating behaves unusually either towards infinity or at an endpoint of the interval. In this context, the integral \[ \int_{1}^{\infty} \frac{1}{x^3} \, dx \] is improper because it has an infinite limit of integration. This means instead of stopping at a finite number, the integration continues indefinitely.

But how do we handle something that keeps going on? We can't calculate it in the usual way. That's where limits help us! They let us explore the behavior of the integral as we extend to infinity. The key is converting our integral into a limit and then checking if this sequence of numbers (integrals at every step to some big value instead of infinity directly) makes sense. If applying a limit gives us a finite number, then our integral is convergent (i.e., it has a valid answer). Otherwise, it's divergent, meaning it doesn't settle down to any number.

Limit Evaluation

When dealing with an improper integral, we transform it into a regular one using limits. Here, we take \[ \int_{1}^{\infty} \frac{1}{x^3} \, dx \] and rewrite it using a limit:

\[ \lim_{b \to \infty} \int_{1}^{b} \frac{1}{x^3} \, dx \]

This step is crucial for evaluating convergence. The idea is to integrate the function over a large but finite interval from 1 to \( b \), and then observe the result as \( b \) approaches infinity.

The limit expression essentially breaks down a complex infinite integral into manageable parts, allowing us to work with finite numbers. Here, you substitute the upper bound as \( b \), integrate over \([1, b]\), and only then check what happens when \( b \) becomes exceedingly large. If the solution doesn't blow up and stays finite, our original improper integral is convergent.

Antiderivative

Finding the antiderivative is a vital part of evaluating integrals. For our problem, we need to compute the antiderivative of \( \frac{1}{x^3} \). An antiderivative is essentially the reverse process of taking a derivative.

For \( \frac{1}{x^3} \), we look for a function whose derivative is \( \frac{1}{x^3} \). Through basic integration rules, we determine this to be:

\[ -\frac{1}{2x^2} \]

Once we have the antiderivative, we evaluate it over our specified limits \(1\) and \(b\).

In this case, integrating gives us \[ \left[-\frac{1}{2x^2}\right]_{1}^{b} \]

When we plug in those bounds, we find the expression simplifies and eventually approach a simple calculation indicating whether the original improper integral converges. Antiderivatives thus simplify our expression for the integral calculus path, revealing more than the description of rate of change, and showing actual cumulative quantities like area under a curve.