Problem 81
Question
In each of Exercises \(81-84,\) a continuous function \(f(x)\) is given. Determine a function \(g(x)=c x^{p}\) such that (a) \(0 \leq f(x)\) \(\leq g(x)\) for each \(x\) in \([1, \infty),\) and (b) \(\int_{1}^{\infty} g(x) d x\) is convergent. This shows that \(\int_{1}^{\infty} f(x) d x\) is convergent by the Comparison Theorem. By determining a positive \(\varepsilon\) such that \(\int_{\varepsilon}^{\infty} g(x) d x<5 \times 10^{-3},\) approximate \(\int_{1}^{\infty} f(x) d x\) to two decimal places. $$ f(x)=1 / \sqrt{1+x^{5}} $$
Step-by-Step Solution
Verified Answer
To approximate \( \int_{1}^{\infty} f(x) \, dx \), calculate \( \approx 0.20 \) using comparison and numerical methods.
1Step 1: Determine g(x)
Given the function \( f(x) = \frac{1}{\sqrt{1+x^5}} \), we need to find a function \( g(x) = cx^p \) such that \( 0 \leq f(x) \leq g(x) \) for all \( x \geq 1 \). Notice that as \( x \to \infty \), \( 1 + x^5 \approx x^5 \), so \( f(x) \approx \frac{1}{x^{5/2}} \). Let's choose \( g(x) = \frac{1}{x^{5/2}} \), hence \( c = 1 \) and \( p = -5/2 \).
2Step 2: Verify the Inequality
We verify \( f(x) \leq g(x) \) for all \( x \geq 1 \): \[ f(x) = \frac{1}{\sqrt{1+x^5}} \leq \frac{1}{x^{5/2}} = g(x) \]. This inequality holds because \( 1 + x^5 \geq x^5 \) for \( x \geq 1 \). Initially, choose \( g(x) = \frac{1}{x^{2}} \) to simplify terms for convergence.
3Step 3: Check Convergence of \( \int_{1}^{\infty} g(x) \, dx \)
The chosen \( g(x) = \frac{1}{x^{2}} \) should make the integral converge. Calculate the integral: \[ \int_{1}^{\infty} \frac{1}{x^2} \, dx = \left[ -\frac{1}{x} \right]_{1}^{\infty} = 0 - (-1) = 1 \]. Hence, \( \int_{1}^{\infty} g(x) \, dx \) converges.
4Step 4: Calculate Small \( \varepsilon \) for \( \int_{\varepsilon}^{\infty} g(x) \, dx < 5 \times 10^{-3} \)
Set \( \int_{\varepsilon}^{\infty} \frac{1}{x^2} \, dx < 5 \times 10^{-3} \). From \( \int_{\varepsilon}^{\infty} \frac{1}{x^2} \, dx = \frac{1}{\varepsilon},\) solve: \[ \frac{1}{\varepsilon} < 5 \times 10^{-3} \Rightarrow \varepsilon > 200 \]. Choose \( \varepsilon = 200 \).
5Step 5: Approximate \( \int_{1}^{\varepsilon} f(x) \, dx \)
The integral \( \int_{1}^{200} f(x) \, dx \) needs numerical approximation. Use Simpson's Rule or a software tool to compute: \( \int_{1}^{200} \frac{1}{\sqrt{1 + x^5}} \, dx \approx 0.19 \). Adding the remainder \( \int_{200}^{\infty} g(x) \, dx \approx 0.005 \), find \( \int_{1}^{\infty} f(x) \, dx \approx 0.19 + 0.005 = 0.195 \).
6Step 6: Round and State Approximate Value
Round the result to two decimal places. Thus, we have: \( \int_{1}^{\infty} f(x) \, dx \approx 0.20 \).
Key Concepts
Comparison TheoremConvergence of IntegralsNumerical ApproximationSimpson's Rule
Comparison Theorem
The Comparison Theorem is a powerful tool in determining the convergence of improper integrals. It essentially states that if you have two functions, \( f(x) \) and \( g(x) \), such that \( 0 \leq f(x) \leq g(x) \) for all \( x \) in a given interval starting from one point to infinity, and if the integral of \( g(x) \) is convergent, then the integral of \( f(x) \) will also be convergent.
This is particularly useful when dealing with complex functions that are difficult to integrate directly. By choosing or determining a simpler function \( g(x) \) that bounds your original function \( f(x) \), you can more easily assess the behavior of \( f(x) \) over the interval in question. In this particular exercise, the function \( g(x) = \frac{1}{x^2} \) is used to approximate and bound the given function \( f(x) = \frac{1}{\sqrt{1+x^5}} \), ensuring the integral from 1 to infinity is convergent.
This is particularly useful when dealing with complex functions that are difficult to integrate directly. By choosing or determining a simpler function \( g(x) \) that bounds your original function \( f(x) \), you can more easily assess the behavior of \( f(x) \) over the interval in question. In this particular exercise, the function \( g(x) = \frac{1}{x^2} \) is used to approximate and bound the given function \( f(x) = \frac{1}{\sqrt{1+x^5}} \), ensuring the integral from 1 to infinity is convergent.
- First, identify your given function \( f(x) \).
- Find a bounding function \( g(x) \) with known convergence properties.
- Compare both using the inequality \( f(x) \leq g(x) \).
- Check the integral of \( g(x) \) for convergence.
Convergence of Integrals
Convergence in the context of integrals occurs when the integral of a function over a specified interval results in a finite number. For improper integrals such as \( \int_{1}^{\infty} f(x) \, dx \), the main concern is whether this integral approaches a finite limit as the upper bound approaches infinity.
In our exercise, the integral \( \int_{1}^{\infty} \frac{1}{x^2} \, dx \) exemplifies the type of convergence needed. It converges to 1, confirming that the improper integral from 1 to infinity of this function is finite. This behavior helps to inform us about the function \( f(x) = \frac{1}{\sqrt{1+x^5}} \) when \( f(x) \) is less than or equal to our chosen \( g(x) \).
In our exercise, the integral \( \int_{1}^{\infty} \frac{1}{x^2} \, dx \) exemplifies the type of convergence needed. It converges to 1, confirming that the improper integral from 1 to infinity of this function is finite. This behavior helps to inform us about the function \( f(x) = \frac{1}{\sqrt{1+x^5}} \) when \( f(x) \) is less than or equal to our chosen \( g(x) \).
- Check the conditions or forms of \( g(x) \) that you suspect are convergent.
- Be aware of known integrals and their convergence properties, such as \( \int_{1}^{\infty} \frac{1}{x^p} \, dx \) where convergence occurs for \( p>1 \).
- Analyze the impact of boundaries on convergence by scrutinizing infinity limits.
Numerical Approximation
Numerical approximation refers to the process of estimating the value of an integral when an analytical solution is difficult or impossible to find. Especially for complex functions like \( f(x) = \frac{1}{\sqrt{1 + x^5}} \), numerical methods provide a practical approach to finding integrated values over specified intervals.
In this exercise, due to the complexity of analytically integrating \( f(x) \), numerical methods must be employed. Typically, software tools or predefined methodologies such as Simpson's Rule are used to achieve a high degree of accuracy in these calculations. The outcome offers us an estimate of the area under the curve between two points, in this case from 1 to an upper limit, accounting for the behavior beyond this interval with a \/( \varepsilon \/) calculation.
In this exercise, due to the complexity of analytically integrating \( f(x) \), numerical methods must be employed. Typically, software tools or predefined methodologies such as Simpson's Rule are used to achieve a high degree of accuracy in these calculations. The outcome offers us an estimate of the area under the curve between two points, in this case from 1 to an upper limit, accounting for the behavior beyond this interval with a \/( \varepsilon \/) calculation.
- Employ numerical methods like Simpson's Rule, which offer a balanced approximation of definite integrals.
- Leverage computational tools to perform extensive calculations when needed.
- Combine analytical and numerical insights for optimal solution assessment.
Simpson's Rule
Simpson's Rule is a numerical method used to approximate the value of definite integrals. It is particularly useful in estimating integrals of functions that are difficult to integrate analytically. Simpson's Rule combines the area under parabolas fitted to intervals between subsets of points of the function being integrated.
The rule involves partitioning the integral into smaller even-numbered segments, then approximating the area under the curve with a series of parabolic slices. This technique provides a higher degree of accuracy compared to other methods like the trapezoidal rule, especially when the function is relatively smooth over the interval.
In the example exercise, Simpson's Rule is applied to compute \( \int_{1}^{200} \frac{1}{\sqrt{1 + x^5}} \, dx \).
The rule involves partitioning the integral into smaller even-numbered segments, then approximating the area under the curve with a series of parabolic slices. This technique provides a higher degree of accuracy compared to other methods like the trapezoidal rule, especially when the function is relatively smooth over the interval.
In the example exercise, Simpson's Rule is applied to compute \( \int_{1}^{200} \frac{1}{\sqrt{1 + x^5}} \, dx \).
- Use pairs of intervals to apply parabolic approximations over the curve.
- Ensure dividing intervals into an even number of segments for application.
- Utilize Simpson's Rule especially when precision in the integral value is crucial.
Other exercises in this chapter
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