Use The Comparison Theorem To Determine Whether The Integral Is Convergent Or Divergent:i. ∫₀^π (sin² X) / (√x) Dxii. ∫₂ [1] (x + Sin X) / (x² - X) Dx Infinity ↩︎
Introduction
The comparison theorem is a powerful tool in calculus used to determine whether an improper integral is convergent or divergent. This theorem allows us to compare the given integral with a known convergent or divergent integral. In this article, we will use the comparison theorem to determine whether the given integrals are convergent or divergent.
Comparison Theorem
The comparison theorem states that if we have two functions f(x) and g(x) such that:
- f(x) ≤ g(x) for all x in the interval [a, ∞)
- ∫a^[∞) g(x) dx is convergent
Then, ∫a^[∞) f(x) dx is also convergent.
On the other hand, if we have two functions f(x) and g(x) such that:
- f(x) ≥ g(x) for all x in the interval [a, ∞)
- ∫a^[∞) g(x) dx is divergent
Then, ∫a^[∞) f(x) dx is also divergent.
Example i: ∫₀^π (sin² x) / (√x) dx
To determine whether the integral ∫₀^π (sin² x) / (√x) dx is convergent or divergent, we can use the comparison theorem.
First, we need to find a known convergent or divergent integral to compare with. Let's consider the integral ∫₀^π (sin x)² dx.
We know that (sin x)² ≤ (sin x)² + (1/√x) for all x in the interval [0, π]. Therefore, we can write:
(sin x)² / (√x) ≤ (sin x)² + (1/√x)
Now, we can compare the given integral with the integral ∫₀^π (sin x)² dx.
Since ∫₀^π (sin x)² dx is convergent, we can conclude that ∫₀^π (sin² x) / (√x) dx is also convergent.
Example ii: ∫₂[1] (x + sin x) / (x² - x) dx
To determine whether the integral ∫₂[2] (x + sin x) / (x² - x) dx is convergent or divergent, we can use the comparison theorem.
First, we need to find a known convergent or divergent integral to compare with. Let's consider the integral ∫₂[3] (x + sin x) / x² dx.
We know that (x + sin x) / (x² - x) ≥ (x + sin x) / x² for all x in the interval [2, ∞). Therefore, we can write:
(x + sin x) / (x² - x) ≥ (x + sin x) / x²
Now, we can compare the given integral with the integral ∫₂[4] (x + sin x) / x² dx.
Since ∫₂[5] (x + sin x) / x² dx is divergent, we can conclude that ∫₂[6] (x + sin x) / (x² - x) dx is also divergent.
Conclusion
In this article, we used the comparison theorem to determine whether the given integrals are convergent or divergent. We compared the given integrals with known convergent or divergent integrals and concluded that the first integral is convergent and the second integral is divergent.
Comparison Theorem: A Powerful Tool in Calculus
The comparison theorem is a powerful tool in calculus used to determine whether an improper integral is convergent or divergent. This theorem allows us to compare the given integral with a known convergent or divergent integral.
Key Takeaways
- The comparison theorem states that if we have two functions f(x) and g(x) such that f(x) ≤ g(x) for all x in the interval [a, ∞) and ∫a^[∞) g(x) dx is convergent, then ∫a^[∞) f(x) dx is also convergent.
- The comparison theorem states that if we have two functions f(x) and g(x) such that f(x) ≥ g(x) for all x in the interval [a, ∞) and ∫a^[∞) g(x) dx is divergent, then ∫a^[∞) f(x) dx is also divergent.
- The comparison theorem can be used to determine whether an improper integral is convergent or divergent by comparing it with a known convergent or divergent integral.
Real-World Applications
The comparison theorem has many real-world applications in fields such as physics, engineering, and economics. For example, it can be used to determine whether a physical system is stable or unstable, or whether a financial investment is likely to be profitable or not.
Future Research Directions
The comparison theorem is a powerful tool in calculus, but there are still many open research directions in this area. For example, researchers are still working to develop new comparison theorems that can be used to determine whether an improper integral is convergent or divergent.
Conclusion
Introduction
The comparison theorem is a powerful tool in calculus used to determine whether an improper integral is convergent or divergent. In our previous article, we used the comparison theorem to determine whether the given integrals are convergent or divergent. In this article, we will answer some frequently asked questions about the comparison theorem.
Q: What is the comparison theorem?
A: The comparison theorem is a mathematical tool used to determine whether an improper integral is convergent or divergent by comparing it with a known convergent or divergent integral.
Q: How do I use the comparison theorem?
A: To use the comparison theorem, you need to find a known convergent or divergent integral to compare with the given integral. Then, you need to determine whether the given integral is less than or greater than the known integral. If the given integral is less than the known integral and the known integral is convergent, then the given integral is also convergent. If the given integral is greater than the known integral and the known integral is divergent, then the given integral is also divergent.
Q: What are some common mistakes to avoid when using the comparison theorem?
A: Some common mistakes to avoid when using the comparison theorem include:
- Not finding a known convergent or divergent integral to compare with
- Not determining whether the given integral is less than or greater than the known integral
- Not considering the limits of integration
- Not considering the properties of the functions involved
Q: Can I use the comparison theorem to determine whether a definite integral is convergent or divergent?
A: Yes, you can use the comparison theorem to determine whether a definite integral is convergent or divergent. However, you need to be careful when choosing the known integral to compare with, as the limits of integration may affect the result.
Q: Can I use the comparison theorem to determine whether an improper integral is convergent or divergent?
A: Yes, you can use the comparison theorem to determine whether an improper integral is convergent or divergent. The comparison theorem can be used to determine whether an improper integral is convergent or divergent by comparing it with a known convergent or divergent integral.
Q: What are some real-world applications of the comparison theorem?
A: The comparison theorem has many real-world applications in fields such as physics, engineering, and economics. For example, it can be used to determine whether a physical system is stable or unstable, or whether a financial investment is likely to be profitable or not.
Q: Can I use the comparison theorem to determine whether a function is continuous or discontinuous?
A: No, the comparison theorem is not used to determine whether a function is continuous or discontinuous. However, the comparison theorem can be used to determine whether an improper integral is convergent or divergent, which can be related to the continuity of the function.
Q: Can I use the comparison theorem to determine whether a function is differentiable or non-differentiable?
: No, the comparison theorem is not used to determine whether a function is differentiable or non-differentiable. However, the comparison theorem can be used to determine whether an improper integral is convergent or divergent, which can be related to the differentiability of the function.
Conclusion
In conclusion, the comparison theorem is a powerful tool in calculus used to determine whether an improper integral is convergent or divergent. We answered some frequently asked questions about the comparison theorem and provided some examples of how to use it. We hope that this article has been helpful in understanding the comparison theorem and its applications.
Comparison Theorem: A Powerful Tool in Calculus
The comparison theorem is a mathematical tool used to determine whether an improper integral is convergent or divergent by comparing it with a known convergent or divergent integral. It is a powerful tool in calculus and has many real-world applications.
Key Takeaways
- The comparison theorem is a mathematical tool used to determine whether an improper integral is convergent or divergent.
- The comparison theorem can be used to determine whether a definite integral is convergent or divergent.
- The comparison theorem can be used to determine whether an improper integral is convergent or divergent.
- The comparison theorem has many real-world applications in fields such as physics, engineering, and economics.
Real-World Applications
The comparison theorem has many real-world applications in fields such as physics, engineering, and economics. For example, it can be used to determine whether a physical system is stable or unstable, or whether a financial investment is likely to be profitable or not.
Future Research Directions
The comparison theorem is a powerful tool in calculus, but there are still many open research directions in this area. For example, researchers are still working to develop new comparison theorems that can be used to determine whether an improper integral is convergent or divergent.
Conclusion
In conclusion, the comparison theorem is a powerful tool in calculus used to determine whether an improper integral is convergent or divergent. We answered some frequently asked questions about the comparison theorem and provided some examples of how to use it. We hope that this article has been helpful in understanding the comparison theorem and its applications.