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Chapter 5: Problem 11

Determine whether each improper integral is convergent or divergent, and find its value if it is convergent. $$ \int_{2}^{\infty} 4 x^{-2} d x $$

Short Answer

Expert verified
The integral is convergent with a value of 2.

Step by step solution

01

Set Up the Integral

The given integral is \( \int_{2}^{\infty} 4x^{-2} \, dx \). This represents an improper integral due to the upper limit being infinite. To evaluate it, we will replace the upper limit with a variable \( b \) and later take the limit as \( b \rightarrow \infty \).
02

Calculate the Integral

First, integrate the function \( 4x^{-2} \). Rewrite the integrand as \( 4x^{-2} = \frac{4}{x^2} \). The integral of \( \frac{4}{x^2} \) is \( -\frac{4}{x} \). So, the antiderivative is \( -\frac{4}{x} \), and the evaluated integral from \( 2 \) to \( b \) is \[ \int_2^b 4x^{-2} \, dx = \left[ -\frac{4}{x} \right]_2^b = -\frac{4}{b} + \frac{4}{2}. \]
03

Evaluate the Limits

Now, we need to find the limit of the evaluated integral as \( b \rightarrow \infty \). The integral becomes \[ \lim_{b \to \infty} \left( -\frac{4}{b} + 2 \right). \] As \( b \) approaches infinity, \( -\frac{4}{b} \) approaches \( 0 \). Thus, the limit becomes \[ \lim_{b \to \infty} \left( 0 + 2 \right) = 2. \]
04

Determine Convergence

Since the limit is finite, the improper integral is convergent. The value of the convergent integral is 2.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Convergence of Integrals
Improper integrals often have infinite limits of integration or discontinuous integrands at bounds. To determine if an integral is convergent, we assess whether the integral approaches a finite limit.
In our example, we look at the integral \( \int_{2}^{\infty} 4x^{-2} \, dx \). The convergence is verified by evaluating the limit of the definite integral as the upper bound approaches infinity. We replaced \( \infty \) with a finite variable \( b \) and then calculated \( \lim_{b \to \infty} \left( -\frac{4}{b} + 2 \right) \).
Since this expression approaches 2, a finite number, the integral converges.To identify convergence in general:
  • Integrals converge when the area under the curve of the function is finite.
  • If the integral approaches infinity or does not stabilize to a number, it diverges.
  • Use limit tests for improper integrals where limits are infinite.
Integration Techniques
Calculating improper integrals efficiently often requires the use of substitution, partial fraction decomposition, or integration by parts. In this case, recognizing the pattern of a power function helped.
For \( \int_{2}^{b} 4x^{-2} \, dx \), rewriting the integrand as \( \frac{4}{x^2} \) is crucial.
Through integration rules, the antiderivative of \( \frac{c}{x^n} \) is \( -\frac{c}{x^{n-1}} + C \), where \( C \) is the constant of integration.
Here, the antiderivative becomes \( -\frac{4}{x} \). Evaluating this from 2 to \( b \) simplifies to \[ \left[ -\frac{4}{x} \right]_2^b = -\frac{4}{b} + 2. \]Some tips on integration techniques:
  • Review common derivatives and integrals to identify patterns easily.
  • Utilize simplification tricks, like algebraic manipulation, to make integration smoother.
  • Choose substitution or parts based on the integrand's setup for simplification.
Limits at Infinity
To evaluate improper integrals with infinite bounds, we need to understand the behavior of the function as it approaches infinity. In this problem, examining \(-\frac{4}{b} + 2\) where \( b \to \infty \) helps us determine the integral's behavior as the variable approaches infinity.
As \( b \) increases without bound, the term \(-\frac{4}{b}\) becomes closer to 0, leaving the expression to simplify to 2.Key concepts for limits at infinity:
  • As a function's denominator grows infinite, such as in \( \frac{1}{x} \), the term diminishes, approaching zero.
  • Constants added to diminishing terms generally remain unaffected by limits.
  • For larger functions, analyzing the highest power of x can predict end behavior.
Thus, understanding limits at infinity allows us to determine whether the function stabilizes, leading to a convergent integral.

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