Proper Integrals: Show Your Work To Answer The Following Questions.Find All Real Numbers $p$ For Which The Integral $\int_1^{\infty} \frac{1}{x^{1-p}} \, Dx$ Is Convergent. (So, For What $p$ Is The Integral Divergent?)

Introduction

In calculus, improper integrals are used to find the area under curves that extend to infinity. These integrals are called improper because they involve limits that are not finite. In this article, we will explore the convergence and divergence of improper integrals of the form $\int_1^{\infty} \frac{1}{x^{1-p}} \, dx$, where $p$ is a real number.

Convergence and Divergence

To determine whether an improper integral converges or diverges, we need to evaluate the limit of the integral as the upper bound approaches infinity. In this case, we have:

$$\int_1^{\infty} \frac{1}{x^{1-p}} \, dx = \lim_{b \to \infty} \int_1^b \frac{1}{x^{1-p}} \, dx$$

Evaluating the Integral

To evaluate the integral, we can use the power rule of integration, which states that:

$$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$$

Applying this rule to our integral, we get:

$$\int \frac{1}{x^{1-p}} \, dx = \int x^{p-1} \, dx = \frac{x^p}{p} + C$$

Evaluating the Limit

Now, we need to evaluate the limit of the integral as the upper bound approaches infinity:

$$\lim_{b \to \infty} \int_1^b \frac{1}{x^{1-p}} \, dx = \lim_{b \to \infty} \left[ \frac{x^p}{p} \right]_1^b$$

Case 1: $p > 0$

If $p > 0$, then the limit exists and is equal to:

$$\lim_{b \to \infty} \left[ \frac{x^p}{p} \right]_1^b = \frac{b^p}{p} - \frac{1}{p}$$

As $b$ approaches infinity, the first term approaches infinity, and the second term approaches zero. Therefore, the limit is infinite, and the integral diverges.

Case 2: $p = 0$

If $p = 0$, then the integral becomes:

$$\int_1^{\infty} \frac{1}{x} \, dx = \lim_{b \to \infty} \int_1^b \frac{1}{x} \, dx$$

Evaluating the integral, we get:

$$\int \frac{1}{x} \, dx = \ln|x| + C$$

Taking the limit as the upper bound approaches infinity, we get:

$$\lim_{b \to \infty} \left[ \ln|x| \right]_1^b = \lim_{b \to \infty} (\ln|b| - \ln|1|)$$

As $b$ approaches infinity, the limit approaches infinity, and the integral diverges.

Case 3: $p < 0$

If $p < 0$, then the integral becomes:

$$\int_1^{\infty} x^{p-1} \, dx = \lim_{b \to \infty} \int_1^b x^{p-1} \, dx$$

Evaluating the integral, we get:

$$\int x^{p-1} \, dx = \frac{x^p}{p} + C$$

Taking the limit as the upper bound approaches infinity, we get:

$$\lim_{b \to \infty} \left[ \frac{x^p}{p} \right]_1^b = \lim_{b \to \infty} \left( \frac{b^p}{p} - \frac{1}{p} \right)$$

As $b$ approaches infinity, the first term approaches zero, and the second term approaches zero. Therefore, the limit exists and is equal to:

$$\lim_{b \to \infty} \left( \frac{b^p}{p} - \frac{1}{p} \right) = -\frac{1}{p}$$

Therefore, the integral converges if $p < 0$.

Conclusion

In conclusion, the improper integral $\int_1^{\infty} \frac{1}{x^{1-p}} \, dx$ converges if $p < 0$ and diverges if $p \geq 0$. This result can be used to determine the convergence and divergence of other improper integrals of the form $\int_1^{\infty} x^{p-1} \, dx$.

References

• [1] Apostol, T. M. (1974). Mathematical Analysis. Addison-Wesley.
• [2] Spivak, M. (1965). Calculus. W.A. Benjamin.

Additional Resources

• [1] Khan Academy. (n.d.). Improper Integrals. Retrieved from https://www.khanacademy.org/math/calculus/sequences-series-calculus/series-and-integrals/v/improper-integrals
• [2] MIT OpenCourseWare. (n.d.). 18.01 Single Variable Calculus. Retrieved from https://ocw.mit.edu/courses/mathematics/18-01-single-variable-calculus-fall-2006/
  Q&A: Improper Integrals ==========================

Q: What is an improper integral?

A: An improper integral is a type of integral that involves limits that are not finite. It is used to find the area under curves that extend to infinity.

Q: How do you evaluate an improper integral?

A: To evaluate an improper integral, you need to evaluate the limit of the integral as the upper bound approaches infinity. This can be done using the power rule of integration and the fundamental theorem of calculus.

Q: What is the power rule of integration?

A: The power rule of integration states that:

$$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$$

Q: What is the fundamental theorem of calculus?

A: The fundamental theorem of calculus states that:

$$\int_a^b f(x) \, dx = F(b) - F(a)$$

where $F(x)$ is the antiderivative of $f(x)$.

Q: How do you determine whether an improper integral converges or diverges?

A: To determine whether an improper integral converges or diverges, you need to evaluate the limit of the integral as the upper bound approaches infinity. If the limit exists and is finite, then the integral converges. If the limit does not exist or is infinite, then the integral diverges.

Q: What is the difference between a convergent and a divergent improper integral?

A: A convergent improper integral is one that has a finite limit as the upper bound approaches infinity. A divergent improper integral is one that has an infinite limit as the upper bound approaches infinity.

Q: Can you give an example of a convergent improper integral?

A: Yes, the integral $\int_1^{\infty} \frac{1}{x^2} \, dx$ is a convergent improper integral. To evaluate it, you can use the power rule of integration and the fundamental theorem of calculus:

$$\int \frac{1}{x^2} \, dx = -\frac{1}{x} + C$$

Taking the limit as the upper bound approaches infinity, you get:

$$\lim_{b \to \infty} \left[ -\frac{1}{x} \right]_1^b = \lim_{b \to \infty} \left( -\frac{1}{b} + 1 \right) = 1$$

Therefore, the integral converges to 1.

Q: Can you give an example of a divergent improper integral?

A: Yes, the integral $\int_1^{\infty} \frac{1}{x} \, dx$ is a divergent improper integral. To evaluate it, you can use the power rule of integration and the fundamental theorem of calculus:

$$\int \frac{1}{x} \, dx = \ln|x| + C$$

Taking the limit as the upper bound approaches infinity, you get:

$$\lim_{b \to \infty} \left[ \ln|x| \right]_1^b = \lim_{b \to \infty} (\ln|b| - \ln|1|) = \infty$$

Therefore, the integral diverges.

Q: What are some common types of improper integrals?

A: Some common types of improper integrals include:

• Infinite integrals: These are integrals that extend to infinity, such as $\int_1^{\infty} \frac{1}{x^2} \, dx$.
• Improper integrals with discontinuities: These are integrals that have discontinuities in the integrand, such as $\int_1^{\infty} \frac{1}{x} \, dx$.
• Improper integrals with infinite limits: These are integrals that have infinite limits, such as $\int_{-\infty}^{\infty} \frac{1}{x^2} \, dx$.

Q: How do you use improper integrals in real-world applications?

A: Improper integrals are used in a variety of real-world applications, including:

• Physics: Improper integrals are used to calculate the work done by a force over an infinite distance, such as the work done by gravity on an object in orbit.
• Engineering: Improper integrals are used to calculate the stress and strain on materials under infinite loads, such as the stress on a beam under infinite tension.
• Economics: Improper integrals are used to calculate the present value of infinite streams of income or expenses, such as the present value of a pension fund.

Q: What are some common mistakes to avoid when working with improper integrals?

A: Some common mistakes to avoid when working with improper integrals include:

• Not checking for convergence: Failing to check whether an improper integral converges or diverges can lead to incorrect results.
• Not using the correct limits: Using the wrong limits can lead to incorrect results.
• Not using the correct antiderivative: Using the wrong antiderivative can lead to incorrect results.

Q: How do you debug improper integrals?

A: Debugging improper integrals involves checking for convergence, using the correct limits, and using the correct antiderivative. It also involves checking for common mistakes, such as not checking for convergence or using the wrong limits.

Q: What are some resources for learning more about improper integrals?

A: Some resources for learning more about improper integrals include:

• Textbooks: There are many textbooks on calculus that cover improper integrals, such as "Calculus" by Michael Spivak and "Calculus: Early Transcendentals" by James Stewart.
• Online resources: There are many online resources that cover improper integrals, such as Khan Academy and MIT OpenCourseWare.
• Practice problems: Practice problems are a great way to learn more about improper integrals. You can find practice problems in textbooks, online resources, and on websites such as Wolfram Alpha.