Problem 5 - Calculus for Biology and Medicine

Question

All the integrals are improper and converge. Explain in each case why the integral is improper, andevaluate each integral. $$ \int_{1}^{\infty} \frac{1}{x^{3 / 2}} d x $$

Step-by-Step Solution

Answer

The improper integral converges to 2.

Step 1: Identify why the integral is improper

The integral \( \int_{1}^{\infty} \frac{1}{x^{3/2}} \, dx \) is considered improper because it is evaluated over an unbounded interval: \([1, \infty)\). The upper limit of integration is infinity, which means the integral extends to an infinitely large value.

Step 2: Set up the limit

To solve an improper integral, we replace the infinite limit with a variable and take the limit as the variable approaches infinity. Set up the integral as follows:\[\int_{1}^{\infty} \frac{1}{x^{3/2}} \, dx = \lim_{b \to \infty} \int_{1}^{b} \frac{1}{x^{3/2}} \, dx.\]

Step 3: Find the antiderivative

The antiderivative of \( \frac{1}{x^{3/2}} \) can be determined by using the power rule for integration. Rewrite \( \frac{1}{x^{3/2}} \) as \( x^{-3/2} \). The antiderivative of \( x^{-3/2} \) is \( -2x^{-1/2} = -\frac{2}{\sqrt{x}} \).

Step 4: Evaluate the definite integral

Apply the limits of integration to the antiderivative:\[\int_{1}^{b} \frac{1}{x^{3/2}} \, dx = \left[ -\frac{2}{\sqrt{x}} \right]_{1}^{b} = -\frac{2}{\sqrt{b}} - \left(-\frac{2}{\sqrt{1}}\right) = \frac{2}{\sqrt{1}} - \frac{2}{\sqrt{b}} = 2 - \frac{2}{\sqrt{b}}.\]

Step 5: Take the limit as b approaches infinity

Evaluate the limit to solve the improper integral:\[\lim_{b \to \infty} \left(2 - \frac{2}{\sqrt{b}}\right) = 2 - \lim_{b \to \infty} \frac{2}{\sqrt{b}}.\]As \( b \rightarrow \infty \), \( \frac{2}{\sqrt{b}} \rightarrow 0 \). Thus, the limit becomes: 2.

Key Concepts

Unbounded Interval

An integral is considered improper when it involves an unbounded interval, which means one of the limits of integration is infinite. In this case, the integral \( \int_{1}^{\infty} \frac{1}{x^{3/2}} \, dx \) has an upper limit of integration approaching infinity. This results in the necessity to find a way to effectively evaluate such integrals, as traditional methods for finite intervals do not directly apply.

Understanding unbounded intervals is crucial, especially when dealing with integrals that stretch indefinitely in one direction. This is common in many problems involving growth rates or decay over infinite time spans. By recognizing the unbounded nature, we know to approach the problem by introducing limits, helping us evaluate these infinite dimensions pragmatically.

Antiderivative

Finding the antiderivative is an essential step in solving integrals, whether they are proper or improper. In our example, we need to find the antiderivative of the function \( \frac{1}{x^{3/2}} \). To do this, we apply the power rule for integration. This involves rewriting \( \frac{1}{x^{3/2}} \) as \( x^{-3/2} \).

Applying the power rule, the antiderivative of \( x^{-3/2} \) becomes \( -2x^{-1/2} \) or equivalently, \( -\frac{2}{\sqrt{x}} \).

Calculating the antiderivative is a way of "undoing" the differentiation, providing us with a family of functions that, when derived, gives back the original function. In this exercise, it sets the stage for evaluating the improper integral across the specified unbounded interval.

Limit of Integration

Evaluating an improper integral involving an infinite limit necessitates using limits. After finding the antiderivative, the next step is to apply the limit of integration to obtain the numerical value. You employ the result of the previous step, substituting it into the bounds of the definite integral.

In this context, for the integral \( \lim_{b \to \infty} \int_{1}^{b} \frac{1}{x^{3/2}} \, dx \), we substitute \( -\frac{2}{\sqrt{x}} \) from step 3 into the limits from 1 to \( b \). The expression becomes \( \left[ -\frac{2}{\sqrt{x}} \right]_{1}^{b} = 2 - \frac{2}{\sqrt{b}} \).

It's crucial to compute \( \lim_{b \to \infty} \left( 2 - \frac{2}{\sqrt{b}} \right) \). As \( b \) heads towards infinity, \( \frac{2}{\sqrt{b}} \) approaches zero, simplifying the entire expression to the value 2. This process effectively "bounds" the otherwise infinite interval through limits, providing a concrete outcome for an abstract concept.