Evaluate The Following Integral Or State That It Diverges.${ \int_{-27}^8 \frac{d X}{\sqrt[3]{x}} } S E L E C T T H E C O R R E C T C H O I C E A N D , I F N E C E S S A R Y , F I L L I N T H E A N S W E R . A . T H E I N T E G R A L C O N V E R G E S , A N D \[ Select The Correct Choice And, If Necessary, Fill In The Answer.A. The Integral Converges, And \[ S E L Ec Tt H Ecorrec T C H O I Ce An D , I F N Ecess A Ry , F I Ll In T H E An S W Er . A . T H E In T E G R A L Co N V Er G Es , An D \[ \int_{-27}^8 \frac{d X}{\sqrt[3]{x}}

Introduction

In this article, we will evaluate the convergence of the given integral, $\int_{-27}^8 \frac{d x}{\sqrt[3]{x}}$. The integral in question involves a cube root in the denominator, which may lead to a potential issue of divergence. Our goal is to determine whether the integral converges or diverges, and if it converges, we will provide the correct value.

Understanding the Integral

The given integral is $\int_{-27}^8 \frac{d x}{\sqrt[3]{x}}$. To begin, let's break down the integral and understand its components. The integral is a definite integral, meaning it has a specific upper and lower bound. In this case, the lower bound is $-27$ and the upper bound is $8$. The integrand is $\frac{1}{\sqrt[3]{x}}$, which can be rewritten as $x^{-\frac{1}{3}}$.

Convergence of the Integral

To determine whether the integral converges or diverges, we need to examine the behavior of the integrand as $x$ approaches the lower and upper bounds. Let's consider the lower bound first. As $x$ approaches $-27$, the value of the integrand approaches $(-27)^{-\frac{1}{3}}$. Since $-27$ is a negative number, the cube root of $-27$ is also negative. However, the absolute value of the cube root of $-27$ is greater than $1$, which means that the integrand approaches infinity as $x$ approaches $-27$.

Now, let's consider the upper bound. As $x$ approaches $8$, the value of the integrand approaches $(8)^{-\frac{1}{3}}$. Since $8$ is a positive number, the cube root of $8$ is also positive. The absolute value of the cube root of $8$ is less than $1$, which means that the integrand approaches $0$ as $x$ approaches $8$.

Conclusion

Based on our analysis, we can conclude that the integral $\int_{-27}^8 \frac{d x}{\sqrt[3]{x}}$ diverges. The integrand approaches infinity as $x$ approaches $-27$, which means that the integral does not converge.

Answer

The correct answer is:

• The integral diverges.

Discussion

The given integral is a classic example of a divergent integral. The presence of the cube root in the denominator leads to a potential issue of divergence, which we have successfully identified. The analysis of the integral's behavior as $x$ approaches the lower and upper bounds has provided valuable insights into the convergence of the integral.

Related Topics

• Convergence of integrals
• Divergence of integrals
• Properties of definite integrals

References

• [1] Calculus, 3rd edition, Michael Spivak
• [2] Calculus, 2nd edition, James Stewart

Additional Resources

• Khan Academy: Convergence of Integrals
• MIT OpenCourseWare: Calculus
• Wolfram Alpha: Integral Convergence
  Evaluating the Convergence of the Integral: Q&A =====================================================

Introduction

In our previous article, we evaluated the convergence of the integral $\int_{-27}^8 \frac{d x}{\sqrt[3]{x}}$ and concluded that it diverges. In this article, we will address some common questions and concerns related to the convergence of the integral.

Q&A

Q: What is the main reason for the divergence of the integral?

A: The main reason for the divergence of the integral is the behavior of the integrand as $x$ approaches the lower bound, $-27$. As $x$ approaches $-27$, the value of the integrand approaches infinity, which means that the integral does not converge.

Q: Why does the integrand approach infinity as $x$ approaches $-27$?

A: The integrand approaches infinity as $x$ approaches $-27$ because the cube root of $-27$ is a negative number, and the absolute value of the cube root of $-27$ is greater than $1$. This means that the integrand grows without bound as $x$ approaches $-27$.

Q: What is the significance of the upper bound, $8$, in the integral?

A: The upper bound, $8$, is significant because it marks the point at which the integrand approaches $0$. However, the integral still diverges because the integrand approaches infinity as $x$ approaches the lower bound, $-27$.

Q: Can the integral be made to converge by changing the upper bound?

A: No, the integral cannot be made to converge by changing the upper bound. The integral diverges because the integrand approaches infinity as $x$ approaches the lower bound, $-27$, regardless of the value of the upper bound.

Q: What are some common mistakes to avoid when evaluating the convergence of an integral?

A: Some common mistakes to avoid when evaluating the convergence of an integral include:

• Failing to consider the behavior of the integrand as $x$ approaches the lower and upper bounds.
• Assuming that the integral converges simply because the integrand approaches $0$ as $x$ approaches the upper bound.
• Failing to recognize that the integral may diverge even if the integrand approaches $0$ as $x$ approaches the upper bound.

Q: How can I determine whether an integral converges or diverges?

A: To determine whether an integral converges or diverges, you should:

• Examine the behavior of the integrand as $x$ approaches the lower and upper bounds.
• Consider the properties of the integrand, such as its growth rate and behavior at infinity.
• Use mathematical techniques, such as the comparison test or the limit comparison test, to determine whether the integral converges or diverges.

Conclusion

In this article, we have addressed some common questions and concerns related to the convergence of the integral $\int_{-27}^8 \frac{d x}{\sqrt[3]{x}}$. We have emphasized the importance of considering the behavior of the integrand as $x$ approaches the lower and upper bounds, and have provided some common mistakes to avoid when evaluating the convergence of an integral.

Related Topics

• Convergence of integrals
• Divergence of integrals
• Properties of definite integrals
• Comparison test
• Limit comparison test

References

• [1] Calculus, 3rd edition, Michael Spivak
• [2] Calculus, 2nd edition, James Stewart
• [3] Calculus: Early Transcendentals, 2nd edition, James Stewart

Additional Resources

• Khan Academy: Convergence of Integrals
• MIT OpenCourseWare: Calculus
• Wolfram Alpha: Integral Convergence