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Chapter 6: Problem 5

For Activities 5 through \(16,\) evaluate the improper integral. $$ \int_{0.36}^{\infty} 9.6 x^{-0.432} d x= $$

Short Answer

Expert verified
The integral diverges to infinity.

Step by step solution

01

Identify the Nature of the Integral

The integral given is an improper integral because the upper limit is infinity. This means we need to evaluate it by taking a limit as the upper limit approaches infinity.
02

Rewrite the Integral Using a Limit

We express the improper integral as a limit: \[ \int_{0.36}^{ ext{鈭瀩} 9.6 x^{-0.432} \, dx = \lim_{b \to \infty} \int_{0.36}^{b} 9.6 x^{-0.432} \, dx \]
03

Find the Antiderivative

To find the antiderivative of the integral, we need to apply the power rule for integration. The integral of \( x^{n} \) with respect to \( x \) is \( \frac{x^{n+1}}{n+1} \). Here, \( n = -0.432 \), so the antiderivative is: \[ 9.6 \cdot \frac{x^{-0.432+1}}{-0.432+1} = \frac{9.6 x^{0.568}}{0.568} \]
04

Simplify the Antiderivative

Simplify the constant factor: \[ \frac{9.6}{0.568} = 16.9014 \]So, the antiderivative becomes: \[ 16.9014 \cdot x^{0.568} \]
05

Evaluate the Definite Integral

Using the antiderivative found, evaluate the definite integral from \( 0.36 \) to \( b \): \[ 16.9014 \cdot \left[ x^{0.568} \right]_{0.36}^{b} = 16.9014 \cdot \left( b^{0.568} - (0.36)^{0.568} \right) \]
06

Take the Limit as \( b \to \infty \)

Evaluate the limit as \( b \) approaches infinity: \[ \lim_{b \to \infty} 16.9014 \cdot \left( b^{0.568} - (0.36)^{0.568} \right) \]Since \( b^{0.568} \to \infty \), the integral diverges because the integral approaches infinity.

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Key Concepts

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

Antiderivative
An antiderivative is a function that reverses the process of differentiation. It's what you find when you "undo" a derivative. When dealing with an integral, particularly indefinite ones, finding an antiderivative is a vital step. In the context of our exercise, we dealt with a specific power of x:
  • The function was of the form \( x^n \), with our \( n \) being equal to -0.432.
  • The power rule for integrals helps us find the antiderivative for such functions. It states that for a function \( x^n \), the antiderivative is \( \frac{x^{n+1}}{n+1} \), given \( n eq -1 \).
  • Applying this rule to our function, the antiderivative became \( \frac{9.6 x^{0.568}}{0.568} \).
Performing these steps may seem complex initially, but with practice, recognizing these forms and finding their antiderivatives becomes a systematic approach. Remember, the antiderivative gives us a new function that shows how much area "accumulates" under a curve as x increases.
Limits in Calculus
Limits form the bedrock of calculus, enabling us to understand behaviors of functions as inputs approach certain values.
  • In the context of improper integrals, limits allow us to handle the challenge of infinity. Since one of our integral's limits was infinity, finding the value of such an integral involves evaluating a limit.
  • We transformed the integral's upper boundary to a variable \( b \) and expressed it in a limit: \( \lim_{b \to \infty} \int_{0.36}^{b} 9.6 x^{-0.432} \, dx \).
  • This method helps us "approach" the infinite value, determining how the integral behaves as it stretches toward infinity.
Mastering limits is crucial for grasping more complex calculus concepts. By determining how functions behave at extremes, we unravel much about their integral properties.
Divergence of Integrals
In calculus, divergence is a crucial concept describing the behavior of integrals that doesn't converge to a finite value. Such integrals "diverge," typically towards infinity.
  • For an improper integral, like our example, this means that even though we perform all integral-reducing calculations, the integral remains boundless.
  • In this exercise, after applying limits and evaluating from \( 0.36 \) to \( b \), as \( b \to \infty \), the expression \( b^{0.568} \) continued to grow without bounds.
  • Thus, the integral diverged since its value reached infinity. Divergent integrals often hint at the underlying growth of the function being integrated.
Understanding divergence helps illuminate the nature of functions represented by such integrals. Recognizing divergence leads to ponder deeper on the function's inherent expansion or the way it stretches over its domain.

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