Chapter 2: Problem 46
Find an upper bound for the error in estimating \(\int_{4}^{5} \frac{1}{(x-1)^{2}} d x\) using the trapezoidal rule with seven subdivisions.
Short Answer
Expert verified
The error bound is approximately 0.000126.
Step by step solution
01
Define the Integral
We are asked to estimate the integral \(\int_{4}^{5}\frac{1}{(x-1)^{2}} dx\) using the trapezoidal rule with seven subdivisions. The interval of integration is from 4 to 5.
02
Determine the Number of Subdivisions
Since the problem specifies 7 subdivisions, let's denote the number of subdivisions as \(n = 7\).
03
Calculate the Step Size
The step size \(h\) is calculated by dividing the interval length by the number of subdivisions. Thus, \(h = \frac{5 - 4}{7} = \frac{1}{7}\).
04
Formula for Error in Trapezoidal Rule
The error \(E_T\) in using the trapezoidal rule for approximating \(\int_{a}^{b} f(x) \, dx\) is bounded by:\[E_T \leq \frac{(b-a)^3}{12n^2} \max_{x \in [a, b]} |f''(x)|\]where \(f(x) = \frac{1}{(x-1)^2}\) in this problem.
05
Calculate the Second Derivative
Calculate the second derivative of \(f(x) = \frac{1}{(x-1)^2}\): \[f'(x) = -\frac{2}{(x-1)^3}\]\[f''(x) = \frac{6}{(x-1)^4}\]
06
Find Maximum of the Second Derivative
Determine the maximum value of \(|f''(x)|\) over the interval \([4, 5]\). Since \(|f''(x)| = \left|\frac{6}{(x-1)^4}\right|\), the maximum value occurs at the left endpoint (where \(x\) is minimum in this interval) because the denominator decreases as \(x\) increases. Therefore, \(\max_{x \in [4, 5]} |f''(x)| = |f''(4)| = \frac{6}{3^4} = \frac{6}{81} = \frac{2}{27}\).
07
Apply the Error Formula
Plugging into the error bound formula, we have:\[E_T \leq \frac{(5-4)^3}{12 \cdot 7^2} \cdot \frac{2}{27}\]\[E_T \leq \frac{1}{12 \times 49} \times \frac{2}{27}\]\[E_T \leq \frac{2}{15876} \approx 0.000126\]
08
Conclude the Error Bound
The upper bound for the error in estimating the integral using the trapezoidal rule with seven subdivisions is approximately 0.000126.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Numerical Integration
Numerical integration is a technique used to estimate the value of a definite integral when finding an exact solution might be difficult or impossible. Instead of computing the integral analytically, we break the area under the curve into smaller sections and approximate the sum of these sections. A common method is the trapezoidal rule, which approximates the integral by dividing the curve into trapezoids rather than rectangles. This approach helps achieve more accuracy by considering the slope of the curve.
The trapezoidal rule sums the areas of trapezoids formed beneath the curve by connecting series of points along the curve, each corresponding to a small interval, or step size. The function's values at the endpoints of these intervals are used, forming the bases of the trapezoids, with the step size acting as their height.
Using more subdivisions, or smaller step sizes, leads to a better approximation, as the trapezoids will fit the curve more closely. In our example, the interval from 4 to 5 is divided into 7 subdivisions, making the step size small enough to improve the approximation while keeping computations manageable.
The trapezoidal rule sums the areas of trapezoids formed beneath the curve by connecting series of points along the curve, each corresponding to a small interval, or step size. The function's values at the endpoints of these intervals are used, forming the bases of the trapezoids, with the step size acting as their height.
Using more subdivisions, or smaller step sizes, leads to a better approximation, as the trapezoids will fit the curve more closely. In our example, the interval from 4 to 5 is divided into 7 subdivisions, making the step size small enough to improve the approximation while keeping computations manageable.
Error Analysis
In numerical integration, understanding error analysis is crucial to ensure that the approximation is reliable. The error in the trapezoidal rule is based on the second derivative of the function being integrated. This is because the error depends greatly on the curvature of the function's graph.
The error formula for the trapezoidal rule is given by:
\[E_T \leq \frac{(b-a)^3}{12n^2} \max_{x \in [a, b]} |f''(x)|\]
where \(n\) is the number of subdivisions, \(b-a\) is the interval length, and \(f''(x)\) is the second derivative of the function being integrated. The highest value of the absolute second derivative within the interval is used to determine the maximum error.
Our analysis revealed that for \(f(x) = \frac{1}{(x-1)^2}\), the maximum of \(|f''(x)|\) in the interval \([4, 5]\) occurs at \(x=4\), which resulted in a very small error bound of approximately 0.000126. This indicates that the trapezoidal rule with seven subdivisions provides a highly accurate estimate for this particular integral.
The error formula for the trapezoidal rule is given by:
\[E_T \leq \frac{(b-a)^3}{12n^2} \max_{x \in [a, b]} |f''(x)|\]
where \(n\) is the number of subdivisions, \(b-a\) is the interval length, and \(f''(x)\) is the second derivative of the function being integrated. The highest value of the absolute second derivative within the interval is used to determine the maximum error.
Our analysis revealed that for \(f(x) = \frac{1}{(x-1)^2}\), the maximum of \(|f''(x)|\) in the interval \([4, 5]\) occurs at \(x=4\), which resulted in a very small error bound of approximately 0.000126. This indicates that the trapezoidal rule with seven subdivisions provides a highly accurate estimate for this particular integral.
Calculus
Calculus is the mathematical study that allows us to understand change and motion, involving concepts of derivatives and integrals. It provides us with the tools to model and predict patterns in natural phenomena. The concept of integration, particularly, is vital for finding areas under curves, volumes, and other related quantities.
The integral in this exercise, \(\int_{4}^{5} \frac{1}{(x-1)^{2}} dx\), is solved approximately here using the trapezoidal rule from numerical integration. The second derivative, \(f''(x)\), plays a vital role in calculating the error bound for this estimation. It informs us about the curvature of the function, which affects the precision of the trapezoidal approximation.
Learning how to differentiate and integrate functions, as well as understand their geometric interpretations, is essential for grasping advanced topics in calculus and its applications across science, engineering, and economics.
The integral in this exercise, \(\int_{4}^{5} \frac{1}{(x-1)^{2}} dx\), is solved approximately here using the trapezoidal rule from numerical integration. The second derivative, \(f''(x)\), plays a vital role in calculating the error bound for this estimation. It informs us about the curvature of the function, which affects the precision of the trapezoidal approximation.
Learning how to differentiate and integrate functions, as well as understand their geometric interpretations, is essential for grasping advanced topics in calculus and its applications across science, engineering, and economics.
