Problem 22 - Calculus for Biology and Medicine

Question

Determine whether each integral is convergent. If the integral is convergent, compute its value. $$ \int_{0}^{4} \frac{1}{x^{1 / 4}} d x $$

Step-by-Step Solution

Answer

The integral is convergent with a value of \( \frac{8 \sqrt{2}}{3} \).

Step 1: Analyze the Integrand

Consider the function \( f(x) = \frac{1}{x^{1/4}} \). This function is defined and continuous on the interval \((0, 4]\). However, at \(x = 0\), it tends towards infinity, which suggests a possible improper integral.

Step 2: Set up the Improper Integral

To deal with the improper behavior at \(x = 0\), set up the integral as a limit: \[ \lim_{a \to 0^+} \int_{a}^{4} \frac{1}{x^{1/4}} \, dx \] This limit will help in examining the convergence at the lower bound.

Step 3: Find the Antiderivative

Calculate the antiderivative of \(\frac{1}{x^{1/4}}\). By using the power rule, we have:\[ \int \frac{1}{x^{1/4}} \, dx = \int x^{-1/4} \, dx = \frac{x^{3/4}}{3/4} + C = \frac{4}{3} x^{3/4} + C. \]

Step 4: Evaluate the Limit of the Antiderivative

Substitute the limits into the antiderivative: \[ \lim_{a \to 0^+} \left( \frac{4}{3} x^{3/4} \right)\Big|_{a}^{4} = \lim_{a \to 0^+} \left( \frac{4}{3} \times 4^{3/4} - \frac{4}{3} \times a^{3/4} \right). \]

Step 5: Calculate the Limit

Evaluate the limit:- Calculate \( \frac{4}{3} \times 4^{3/4} = \frac{4}{3} \times (2^3)^{1/4} = \frac{4}{3} \times 2^{3/4} \).- For the term \( \frac{4}{3} \times a^{3/4} \), as \( a \to 0^+ \), this becomes 0.- Simplify the expression: The final result is \( \frac{4 \cdot 2^{3/2}}{3} = \frac{8 \sqrt{2}}{3} \).

Step 6: Conclusion

Since the limit exists and is finite, the integral is convergent. The value of the integral is \( \frac{8 \sqrt{2}}{3} \).

Key Concepts

Convergence of Integrals

Convergence of integrals refers to whether the value of an integral approaches a finite limit or not. Improper integrals are a type of integrals where one or both of the limits are infinite, or where the integrand approaches infinity within the limits. To determine if an improper integral is convergent, we set up the integral as a limit and check if it results in a finite number.
In the given exercise, the integral \( \int_{0}^{4} \frac{1}{x^{1/4}} \, dx \) is improper because the integrand \( \frac{1}{x^{1/4}} \) goes to infinity as \( x \) approaches 0. We handle this by expressing the integral as \( \lim_{a \to 0^+} \int_{a}^{4} \frac{1}{x^{1/4}} \, dx \), thereby transforming the problem into evaluating a limit.
This approach helps in determining whether the integral converges i.e., has a finite value. If the computed limit is a definite number, the integral is convergent; otherwise, it's divergent. In this particular case, after evaluation, it is found that the integral converges and holds a value of \( \frac{8 \sqrt{2}}{3} \). This demonstrates that the function doesn't increase to infinity rapidly enough within the interval \([0,4]\) to make the entire area under the curve infinite.

Antiderivative

Finding the antiderivative is a crucial step in solving integrals, especially for evaluating definite integrals. An antiderivative of a function is an operation where we identify a function whose derivative returns the original function. In simpler terms, it's the reverse process of differentiation.

When dealing with integrals such as the one given in the exercise, \( \frac{1}{x^{1/4}} \), computing its antiderivative is essential to find the value of the integral. The antiderivative of \( x^{-1/4} \) is found using the basic integration techniques. By employing the power rule for integration, the antiderivative is calculated as:

• Rewriting the function as \( x^{-1/4} \).
• Applying the power rule \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), which gives us \( \frac{x^{3/4}}{3/4} + C \).
• Simplifying the expression leads to \( \frac{4}{3} x^{3/4} + C \), where \( C \) is the constant of integration.

By calculating this antiderivative, we can then evaluate the definite integral by substituting the given limits and compute the specific value of the integral, hence determining its convergence.

Power Rule

The power rule is a fundamental principle in calculus for both differentiation and integration. It significantly simplifies the process of solving problems involving polynomials and power functions.

When using the power rule for integration, the principle states that for any real number \( n eq -1 \), the integral of \( x^n \) with respect to \( x \) is given by:
\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]
Here, \( C \) is the constant of integration that appears because integration is the inverse operation of differentiation, and thus the derivative of a constant is zero.
In the context of the given exercise, application of the power rule is straightforward. We are tasked with integrating \( \frac{1}{x^{1/4}} \), which is rewritten as \( x^{-1/4} \). The power rule helps in finding the antiderivative as \( \frac{4}{3} x^{3/4} + C \), allowing one to apply the evaluated limits thereafter. This rule is particularly valuable for polynomials and provides a quick shortcut for finding antiderivatives, playing a pivotal role in determining both the convergence and the eventual value of definite integrals, as demonstrated in our solution.