/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 31 The integrals converge. Evaluate... [FREE SOLUTION] | 91影视

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Chapter 8: Problem 31

The integrals converge. Evaluate the integrals without using tables. $$\int_{-1}^{4} \frac{d x}{\sqrt{|x|}}$$

Short Answer

Expert verified
The integral evaluates to 6.

Step by step solution

01

Analyze the Integrand and Interval

The integral given is \( \int_{-1}^{4} \frac{dx}{\sqrt{|x|}} \). The integrand involves an absolute value, i.e., \( |x| \), which affects the domain. For \( x < 0 \), \( |x| = -x \); for \( x \geq 0 \), \( |x| = x \). The interval is \([-1, 4]\), thus we need to split at \( x = 0 \).
02

Split the Integral

The integral is split into two parts due to the absolute value at \( |x| \):\[\int_{-1}^{0} \frac{dx}{\sqrt{-x}} + \int_{0}^{4} \frac{dx}{\sqrt{x}}.\]
03

Evaluate the Integral from -1 to 0

Let's evaluate \( \int_{-1}^{0} \frac{dx}{\sqrt{-x}} \). Substitute \( u = -x \), so \( du = -dx \) or \( dx = -du \). The limits change from \( x = -1 \) to \( u = 1 \) and \( x = 0 \) to \( u = 0 \), so:\[-\int_{1}^{0} \frac{du}{\sqrt{u}} = \int_{0}^{1} u^{-1/2} du.\]This becomes:\[= [2u^{1/2}]_{0}^{1} = 2 \times 1 - 2 \times 0 = 2.\]
04

Evaluate the Integral from 0 to 4

Now evaluate \( \int_{0}^{4} \frac{dx}{\sqrt{x}} \), which is \( \int_{0}^{4} x^{-1/2} dx \).Calculate:\[= [2x^{1/2}]_{0}^{4} = 2 \times 2 - 2 \times 0 = 4.\]
05

Combine the Results

Add the results from the two integrals to find the total integral over \([-1, 4]\):\[2 + 4 = 6.\]
06

Conclusion: Final Evaluation

The integral evaluates to 6 as the absolute values were handled correctly and the parts were combined after calculating each separately.

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Key Concepts

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

Absolute Value in Calculus
In calculus, absolute value expressions like \(|x|\) affect how functions are evaluated, particularly within integrals. Absolute value is defined piecewise for an expression based on the sign of the input.
For any real number \(x\), the absolute value \(|x|\) is equal to \(x\) if \(x \geq 0\) and \(-x\) if \(x < 0\). This impacts domain considerations, especially when integrating over intervals that cross zero.

### How It Affects Integration
  • Integrals involving \(|x|\) often require breaking the integral into parts where the absolute value function is constant.
  • This typically involves determining if \(x\) is positive or negative within subintervals of the original range.
  • Each piece of the split integral can then be computed separately, treating \(|x|\) appropriately in each case.
In our example, the absolute value influenced the approach by necessitating a split at \(x = 0\), dividing the integral into two separate parts to handle \(x\) both less than and greater than zero properly.
Integral Convergence
Integral convergence refers to whether an integral has a finite result as it approaches the bounds of integration. In definite integrals, we seek this finite result over a specified interval.
An integral is said to converge if its limit exists and is finite 鈥� in our case, we ensured convergence by analyzing the behavior of the integrand within each subinterval induced by the absolute value.

### Understanding Convergence in Terms of This Integral
  • Initial observations of the integral from \(-1\) to \(4\) revealed expressions involving \(|x|\) which influence convergence by adjusting the domain.
  • Evaluating the integrals after splitting, considered the effect of the indefinite behavior at \(x = 0\) and near the negative boundary \(-1\).
  • Both integrals in this scenario, \(\int_{-1}^{0}\) and \(\int_{0}^{4}\), converged to finite values, verifying the convergence concept in the context of this problem.
Piecewise Integration
Piecewise integration is a strategy used when functions are not continuous or have different expressions over different intervals. This method divides the integral into parts where standard calculus rules apply separately.
The exercise under consideration had an integrand dependent on \(|x|\), which calls for piecewise integration due to the change in expression at \(x=0\).

### How Piecewise Integration Worked Here
  • The original integral was split at the point of discontinuity of the absolute value, making the intervals \([-1, 0]\) and \([0, 4]\) manageable.
  • Each piece was treated as a standard integral after adjusting the expression for \(|x|\) in each range.
  • This allows for separate evaluation of each integral before combining their results, \(2\) and \(4\), into the final total, \(6\).
Piecewise integration simplifies complex problems by reducing them into smaller, more familiar contexts where straightforward integration rules can be applied.

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Most popular questions from this chapter

Failure time The time between failures of a photocopier is exponentially distributed. Half of the copiers at a university require service during the first 2 years of operations. If the university purchased 150 copiers, how many do you expect to require service during the first year of their operation?

Your metal fabrication company is bidding for a contract to make sheets of corrugated iron roofing like the one shown here. The cross-sections of the corrugated sheets are to conform to the curve $$ y=\sin \frac{3 \pi}{20} x, \quad 0 \leq x \leq 20 \mathrm{in.} $$ If the roofing is to be stamped from flat sheets by a process that does not stretch the material, how wide should the original material be? To find out, use numerical integration to approximate the length of the sine curve to two decimal places.

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Use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer. $$\int_{1}^{\infty} \frac{e^{x}}{x} d x$$
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