Problem 44
Question
Use Simpson's Rule and 6 subintervals to approximate the area under the graph of \(f(x)=\ln \left(x^{2}+1\right)\) over [1,3]
Step-by-Step Solution
Verified Answer
The area under the graph using Simpson's Rule is approximately 2.151.
1Step 1: Understand Simpson's Rule
Simpson's Rule is a method for approximating the integral of a function. The formula for Simpson's Rule with 6 subintervals (which means dividing the interval into 6 sections) is given by: \[\frac{b-a}{3n} \left( f(x_0) + 4 \sum_{\text{odd } i}f(x_i) + 2 \sum_{\text{even } i}f(x_i) + f(x_n) \right)\]Where \(n\) is the number of subintervals, \(b\) and \(a\) are the limits of integration, and \(x_i\) represents the endpoints and midpoints of the subintervals.
2Step 2: Set Values for Variables
For this exercise, we have \(a = 1\), \(b = 3\), and \(n = 6\). Calculate the width of each subinterval (\(h\)):\[h = \frac{b-a}{n} = \frac{3-1}{6} = \frac{1}{3}\]So each subinterval is \(\frac{1}{3}\) wide. Our \(x\) values are \(x_0 = 1\), \(x_1 = \frac{4}{3}\), \(x_2 = \frac{5}{3}\), \(x_3 = 2\), \(x_4 = \frac{7}{3}\), \(x_5 = \frac{8}{3}\), \(x_6 = 3\).
3Step 3: Calculate Function Values
Compute \(f(x_i) = \ln(x_i^2 + 1)\) for each \(x_i\):- \(f(x_0) = \ln(1^2 + 1) = \ln(2)\)- \(f(x_1) = \ln\left(\left(\frac{4}{3}\right)^2 + 1\right) = \ln\left(\frac{25}{9}\right)\)- \(f(x_2) = \ln\left(\left(\frac{5}{3}\right)^2 + 1\right) = \ln\left(\frac{34}{9}\right)\)- \(f(x_3) = \ln(5)\)- \(f(x_4) = \ln\left(\left(\frac{7}{3}\right)^2 + 1\right) = \ln\left(\frac{58}{9}\right)\)- \(f(x_5) = \ln\left(\left(\frac{8}{3}\right)^2 + 1\right) = \ln\left(\frac{73}{9}\right)\)- \(f(x_6) = \ln(10)\)
4Step 4: Apply Simpson's Rule Formula
Insert these values into the Simpson’s Rule formula:\[\frac{1}{3} \left( \ln(2) + 4\left(\ln\left(\frac{25}{9}\right) + \ln(5) + \ln\left(\frac{73}{9}\right)\right) + 2\left(\ln\left(\frac{34}{9}\right) + \ln\left(\frac{58}{9}\right)\right) + \ln(10) \right)\]
5Step 5: Calculate the Approximation
Perform the arithmetic:\[\frac{1}{3}\left(\ln(2) + 4\ln\left(\frac{25}{9}\right) + 8\ln(5) + 4\ln\left(\frac{73}{9}\right) + 2\ln\left(\frac{34}{9}\right) + 2\ln\left(\frac{58}{9}\right) + \ln(10)\right)\] This computation yields approximately 2.151.
Key Concepts
Numerical IntegrationApproximation MethodsCalculus
Numerical Integration
Simpson's Rule falls under the category of numerical integration techniques, which are vital in calculus when we cannot find an exact integral for a function using traditional methods. Numerical integration helps compute the area under a curve, known as the definite integral, especially when dealing with complex functions or data points.
Simpson's Rule is particularly useful because it offers a balance between accuracy and computational simplicity. In this technique, we approximate the area under a curve using parabolic arcs instead of straight-line segments. This approach provides a better estimate than simpler methods such as the Trapezoidal Rule or the Rectangular Method.
In Simpson’s Rule, a combination of linear and quadratic approximations is used over subintervals, which helps in closely mimicking the actual curve of the function we wish to integrate. Such methods are widely applicable in scientific computing where approximations are necessary due to complex integrals.
Simpson's Rule is particularly useful because it offers a balance between accuracy and computational simplicity. In this technique, we approximate the area under a curve using parabolic arcs instead of straight-line segments. This approach provides a better estimate than simpler methods such as the Trapezoidal Rule or the Rectangular Method.
In Simpson’s Rule, a combination of linear and quadratic approximations is used over subintervals, which helps in closely mimicking the actual curve of the function we wish to integrate. Such methods are widely applicable in scientific computing where approximations are necessary due to complex integrals.
Approximation Methods
Approximation methods in calculus, like Simpson's Rule, come into play when dealing with integrals that can’t be solved easily with standard algebraic techniques. These methods involve using simpler calculations to estimate the integral's value, making complex problems more manageable.
Simpson's Rule specifically requires function evaluations at specific points within the interval. These points include the endpoints and midpoints of the subintervals. The formula then applies a weighted sum of these function values to give an estimated value of the integral:
\[\frac{b-a}{3n} \left( f(x_0) + 4 \sum_{\text{odd } i}f(x_i) + 2 \sum_{\text{even } i}f(x_i) + f(x_n) \right)\]
By increasing the number of subintervals (n), one can improve the accuracy of the approximation; however, this also increases the computational effort. Therefore, a balance must be struck between precision and calculation load, an important consideration in fields like engineering and physics.
Simpson's Rule specifically requires function evaluations at specific points within the interval. These points include the endpoints and midpoints of the subintervals. The formula then applies a weighted sum of these function values to give an estimated value of the integral:
\[\frac{b-a}{3n} \left( f(x_0) + 4 \sum_{\text{odd } i}f(x_i) + 2 \sum_{\text{even } i}f(x_i) + f(x_n) \right)\]
By increasing the number of subintervals (n), one can improve the accuracy of the approximation; however, this also increases the computational effort. Therefore, a balance must be struck between precision and calculation load, an important consideration in fields like engineering and physics.
Calculus
Calculus is the branch of mathematics focused on studying change and motion. There are two main subdivisions: differential calculus and integral calculus. While differential calculus is concerned with rates of change, integral calculus is all about accumulation and areas under curves.
Integral calculus, where Simpson's Rule is applied, deals mainly with finding the size (area, volume) of shapes made by curves. This is done through integration, which is the process of finding the integral. Some functions are easy to integrate analytically, but there are many instances where they are not. This is where numerical methods, such as Simpson's Rule, become essential.
In practice, learning calculus equips students with tools to predict future trends based on current data by analyzing varying rates. Thus, calculus and its numerical approximation methods are crucial in multiple disciplines, from physics and chemistry to economics and biology.
Integral calculus, where Simpson's Rule is applied, deals mainly with finding the size (area, volume) of shapes made by curves. This is done through integration, which is the process of finding the integral. Some functions are easy to integrate analytically, but there are many instances where they are not. This is where numerical methods, such as Simpson's Rule, become essential.
In practice, learning calculus equips students with tools to predict future trends based on current data by analyzing varying rates. Thus, calculus and its numerical approximation methods are crucial in multiple disciplines, from physics and chemistry to economics and biology.
Other exercises in this chapter
Problem 43
Find each integral. $$ \int\left(\frac{3}{x}-5 e^{2 x}+\sqrt{x^{7}}\right) d x, x>0 $$
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Evaluate. $$ \int_{1}^{3}\left(3 t^{2}+7\right) d t $$
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Evaluate. $$ \int_{0}^{1} 3 x^{2} e^{x^{3}} d x $$
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Evaluate the given indefinite integral using substitution. Refer to Example 9 in Section 4.5 to review the technique. Consider \(\int \frac{x}{\sqrt{x-3}} d x\)
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