Problem 6 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

The integral is improper due to the upper limit being infinity and evaluates to 2.

Step 1: Identify Why the Integral is Improper

The integral is improper because it has an infinite limit of integration. Specifically, the upper limit is infinity, which makes the integral improper.

Step 2: Rewrite the Integral as a Limit

We redefine the integral to handle the infinity by writing it as a limit: \[ \int_{1}^{ ext{ oop{name}}} \frac{1}{x^{3/2}} \, dx = \lim_{b \to \infty} \int_{1}^{b} \frac{1}{x^{3/2}} \, dx \]

Step 3: Find the Antiderivative

The antiderivative of \( \frac{1}{x^{3/2}} \) is found using the formula \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \). For \( n = -3/2 \), the antiderivative is \[ \int x^{-3/2} \, dx = \frac{x^{-1/2}}{-1/2} = -2x^{-1/2}. \]Thus,\[ \int \frac{1}{x^{3/2}} \, dx = -2x^{-1/2}. \]

Step 4: Evaluate the Definite Integral

Using our antiderivative, we evaluate the definite integral from 1 to \( b \): \[ \int_{1}^{b} \frac{1}{x^{3/2}} \, dx = \left[-2x^{-1/2}\right]_{1}^{b} = -2b^{-1/2} + 2 \cdot 1^{-1/2} \]This simplifies to \[ 2 - \frac{2}{\sqrt{b}}. \]

Step 5: Take the Limit as b Approaches Infinity

We calculate \( \lim_{b \to \infty} \left(2 - \frac{2}{\sqrt{b}}\right) \). As \( b \to \infty \), \( \frac{2}{\sqrt{b}} \to 0 \) because the denominator grows without bound. Therefore, the limit is 2.

Key Concepts

Limit Evaluation

Evaluating limits in calculus is a fundamental technique, particularly crucial when dealing with improper integrals. Improper integrals often have limits that extend to infinity, making direct evaluation impossible. Instead, we use limits to manage this:

• We replace the infinite limit with a variable, such as \( b \), that approaches infinity.
• For example, the integral originally extends from 1 to infinity. We rewrite it as \( \lim_{b \to \infty} \int_{1}^{b} \frac{1}{x^{3/2}} \, dx \).
• This transformation allows us to handle the infinite range by evaluating the integral over a finite interval and then observing the behavior as the interval grows.

By evaluating the integral from 1 to \( b \), and subsequently taking the limit as \( b \) approaches infinity, we can effectively compute an integral that initially seems unsolvable. The limit simplifies the problem and helps in identifying convergence or divergence of the integral.

Antiderivative

An antiderivative is a function whose derivative is the given function. In the context of improper integrals, finding the antiderivative is essential for evaluating the integral. For our specific problem:

• We need the antiderivative of \( \frac{1}{x^{3/2}} \), which can be calculated using the power rule for integration.
• The power rule states: \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), where \( C \) is the constant of integration.
• For \( n = -3/2 \), this becomes \( \int x^{-3/2} \, dx = \frac{x^{-1/2}}{-1/2} = -2x^{-1/2} + C \).

The antiderivative \( -2x^{-1/2} \) allows us to evaluate the definite integral by substituting the bounds into the expression. It is crucial to understand how to derive and apply antiderivatives since they transform the integral into a solvable form.

Convergence of Integrals

The convergence of an integral is about determining whether the integral has a finite value. For improper integrals, which extend to infinity, checking convergence involves examining the behavior of the function as the integration limits stretch:

• We determine convergence by evaluating the limit of the definite integral as the upper bound approaches infinity.
• In the solution, \( \lim_{b \to \infty} \left(2 - \frac{2}{\sqrt{b}}\right) \) was computed.
• Here, \( \frac{2}{\sqrt{b}} \) diminishes to zero as \( b \) grows, signifying that the expression approaches 2.

Thus, the integral converges to 2, indicating that the area under the curve \( \frac{1}{x^{3/2}} \) from 1 to infinity is finite. This concept is key when studying integrals over infinite intervals and helps distinguish between finite and infinite outcomes in calculus.