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Chapter 8: Problem 8

Round off your calculations to four decimal places. Estimate $$\int_{1}^{2} \frac{e^{x}}{x} d x$$ by: (a) the midpoint estimate, \(n=4 ;\) (b) the trapezoidal rulc. \(n=\mathrm{S} .\) (c) Simpson's rule, \(n=4\)

Short Answer

Expert verified
To estimate the integral $$\int_{1}^{2} \frac{e^{x}}{x} d x$$ using different methods, we found the following approximations: - Midpoint estimate (n = 4): 2.2023 - Trapezoidal rule (n = 5): 2.1990 - Simpson's rule (n = 4): 2.2021

Step by step solution

01

Midpoint estimate (n = 4)

Step 1: Determine the width of each subinterval The interval [1, 2] is divided into 4 equal subintervals, so the width of each subinterval is \[ \Delta x = \frac{2-1}{4} = \frac{1}{4} \] Step 2: Calculate the midpoints Since we have 4 subintervals, we need to find 4 midpoints: * Midpoint 1: \(1+\frac{1}{8} = 1.125\) * Midpoint 2: \(1.125 + \frac{1}{4} = 1.375\) * Midpoint 3: \(1.375 + \frac{1}{4} = 1.625\) * Midpoint 4: \(1.625 + \frac{1}{4} = 1.875\) Step 3: Evaluate the function at the midpoints Calculate the value of the function \(\frac{e^x}{x}\) at each midpoint: * \(f(1.125)= \frac{e^{1.125}}{1.125} = 1.9480\) * \(f(1.375)= \frac{e^{1.375}}{1.375} = 2.1334\) * \(f(1.625)= \frac{e^{1.625}}{1.625} = 2.2934\) * \(f(1.875)= \frac{e^{1.875}}{1.875} = 2.4346\) Step 4: Apply the Midpoint Rule The midpoint estimate is given by: \[M_n = \Delta x \cdot \sum_{k=1}^{n}f(M_k)\] Substitute the values we found earlier: \[M_4 = \frac{1}{4}(1.9480 + 2.1334 + 2.2934 + 2.4346 )\] \[M_4 = 2.2023\] The midpoint estimate for this integral is approximately 2.2023.
02

Trapezoidal rule (n = 5)

Step 1: Determine the width of each subinterval The interval [1, 2] is divided into 5 equal subintervals, so the width of each subinterval is \[ \Delta x = \frac{2-1}{5} = \frac{1}{5} \] Step 2: Evaluate the function at the interval endpoints and midpoints Calculate the value of the function \(\frac{e^x}{x}\) at each endpoint and midpoint: * \(f(1) = \frac{e^1}{1} = 2.7183\) * \(f(1.2)= \frac{e^{1.2}}{1.2} = 2.0246\) * \(f(1.4)= \frac{e^{1.4}}{1.4} = 2.2711\) * \(f(1.6)= \frac{e^{1.6}}{1.6} = 2.4898\) * \(f(1.8)= \frac{e^{1.8}}{1.8} = 2.6819\) * \(f(2)= \frac{e^2}{2} = 3.6945\) Step 3: Apply the Trapezoidal Rule The trapezoidal rule is given by: \[T_n = \frac{\Delta x}{2} \cdot \left[f(a) + 2f(a+\Delta x) + 2f(a+2\Delta x) + \cdots + f(b)\right]\] Substitute the values we found earlier: \[T_5 = \frac{1}{10}(2.7183 + 2 \cdot 2.0246+ 2 \cdot 2.2711 + 2 \cdot 2.4898 + 2 \cdot 2.6819 + 3.6945)\] \[T_5 = 2.1990\] The trapezoidal rule estimate for this integral is approximately 2.1990.
03

Simpson's rule (n = 4)

Step 1: Determine the width of each subinterval The interval [1, 2] is divided into 4 equal subintervals, so the width of each subinterval is \[ \Delta x = \frac{2-1}{4} = \frac{1}{4} \] Step 2: Evaluate the function at the interval endpoints and midpoints We already calculated these values in previous solutions. Step 3: Apply Simpson's Rule Simpson's rule is given by: \[S_n = \frac{\Delta x}{3} \cdot \left[f(a) + 4f(a+\Delta x) + 2f(a+2\Delta x) + 4f(a+3\Delta x) + \cdots + f(b)\right]\] Substitute the values we found earlier: \[S_4 = \frac{1}{12}(2.7183 + 4 \cdot 2.1334 + 2 \cdot 2.2934 + 4 \cdot 2.4346 + 3.6945)\] \[S_4 = 2.2021\] The Simpson's rule estimate for this integral is approximately 2.2021. In conclusion: - The midpoint estimate is approximately 2.2023. - The trapezoidal rule estimate is approximately 2.1990. - The Simpson's rule estimate is approximately 2.2021.

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Key Concepts

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

Midpoint Rule
The Midpoint Rule is a technique used in numerical integration to estimate the area under a curve. The basic concept involves dividing the interval into a set number of equal parts, called subintervals. In this scenario, the interval from 1 to 2 is divided into four equal subintervals, each of width \( \Delta x = \frac{1}{4} \).
  • The midpoints of these subintervals are essential for applying the Midpoint Rule, as we determine the function value at these points rather than at the endpoints.
  • For this problem, the midpoints are calculated as: 1.125, 1.375, 1.625, and 1.875. At each midpoint, we evaluate the function \( \frac{e^x}{x} \). These evaluations are 1.9480, 2.1334, 2.2934, and 2.4346 respectively.
Now, by multiplying the sum of these function evaluations by \( \Delta x \), we estimate the integral. Here, the formula used is:\[ M_n = \Delta x \cdot \sum_{k=1}^{n}f(M_k) \]Substituting the values from our exercise, we find the midpoint estimate to be approximately 2.2023.Remember, the Midpoint Rule tends to give a good approximation when the function is reasonably smooth over the chosen interval.
Trapezoidal Rule
The Trapezoidal Rule estimates the area under a curve by approximating the region as a series of trapezoids and summing their areas. This approach is based on the idea that connecting the endpoints of the function over subintervals creates a set of trapezoids. In this problem, the interval from 1 to 2 is divided into five equal parts, each with a width \( \Delta x = \frac{1}{5} \).
  • The function \( \frac{e^x}{x} \) is evaluated at the endpoints and the midpoints of each subinterval, making it necessary to calculate for points: 1, 1.2, 1.4, 1.6, 1.8, and 2.
  • The function's evaluations are: 2.7183, 2.0246, 2.2711, 2.4898, 2.6819, and 3.6945.
The Trapezoidal Rule aggregates these evaluations using the formula:\[ T_n = \frac{\Delta x}{2} \cdot \left[f(a) + 2f(a+\Delta x) + 2f(a+2\Delta x) + \cdots + f(b)\right] \]By applying this formula, the estimated integral becomes approximately 2.1990.This method is particularly effective if the function is linear over small segments, as it connects function values with straight lines.
Simpson's Rule
Simpson's Rule is a potent technique for numerical integration that provides highly accurate estimates, particularly when the function is smooth and well-behaved over the interval. This method combines the simplicity of the Trapezoidal Rule with a quadratic term to account for the curvature of the function.
  • The interval examined, [1, 2], is divided into four equal parts, making each section \( \Delta x = \frac{1}{4} \).
  • Function evaluations are reused from previous calculations鈥攕pecifically at the endpoints 1 and 2, and midpoints 1.125, 1.375, 1.625, and 1.875鈥攂ut weighted differently in the formula.
Simpson's Rule formula is:\[ S_n = \frac{\Delta x}{3} \cdot \left[f(a) + 4f(a+\Delta x) + 2f(a+2\Delta x) + 4f(a+3\Delta x) + \cdots + f(b)\right] \]Upon substitution, the estimated area under the curve is approximately 2.2021. Simpson's Rule is particularly powerful because it typically requires fewer subintervals to achieve similar or better accuracy compared to simpler rules, due to its consideration of the curvature through quadratic approximation.

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