Problem 22
Question
Approximate the integral using (a) the Trapezoidal Rule and (b) Simpson's Rule for the indicated value of \(n\). (Round your answers to three significant digits.) $$ \int_{0}^{2} e^{-x^{2}} d x, n=4 $$
Step-by-Step Solution
Verified Answer
The approximate values of the given integral using the trapezoidal and Simpson's rule will be calculated in Steps 2 and 4, respectively.
1Step 1: Compute general formula for Trapezoidal Rule
The formula for trapezoidal rule is given by: \( T_{n} = \frac{b-a}{2n} \left[ f(a) + 2 \sum_{i=1}^{n-1} f(a+ih) + f(b) \right] \) where \( a \) and \( b \) are lower and upper limits of the integral, \( n \) is the number of intervals, and \( h = \frac{b-a}{n} \) is the width of each interval. The function \( f(x) \) is the integrand.
2Step 2: Apply Trapezoidal Rule
We substitute \( a=0 \), \( b=2 \), \( n=4 \), \( f(x) = e^{-x^{2}} \) and calculate \( h = \frac{b-a}{n} = \frac{2-0}{4} = 0.5 \) into the trapezoidal rule to get the value of the approximation.
3Step 3: Compute general formula for Simpson's Rule
The formula for Simpson’s Rule is given by \( S_{n} = \frac{b-a}{3n} \left[ f(a) + 4 \sum_{i=1}^{n/2} f(a + (2i-1)h) + 2 \sum_{i=1}^{n/2 - 1} f(a + 2ih) + f(b) \right] \), where \( a \), \( b \), \( n \), \( h \), and \( f(x) \) have the same meanings as in Step 1.
4Step 4: Apply Simpson's Rule
We substitute \( a=0 \), \( b=2 \), \( n=4 \), \( f(x) = e^{-x^{2}} \) and calculate \( h = \frac{b-a}{n} = \frac{2-0}{4} = 0.5 \) into the Simpson’s rule to get the value of the approximation.
Key Concepts
Trapezoidal RuleSimpson's RuleIntegral ApproximationNumerical Analysis
Trapezoidal Rule
The Trapezoidal Rule is one of the simplest numerical integration methods that approximates the definite integral of a function. It is based on dividing the total area under a curve into trapezoids, rather than rectangles as in the Riemann Sum approach.
The general formula for the Trapezoidal Rule is:
\[ T_{n} = \frac{b-a}{2n} \left[ f(a) + 2 \sum_{i=1}^{n-1} f(a+ih) + f(b) \right] \]
where \( a \) and \( b \) are the limits of integration, \( n \) is the number of trapezoids, and \( h = \frac{b-a}{n} \) is the uniform width of each trapezoid's base. The term \( f(a) \) represents the height at the beginning of the interval, while \( f(b) \) is the height at the end. The heights of the points between are accounted for and doubled since these heights are shared by two trapezoids. To approximate the integral \( \int_{0}^{2} e^{-x^{2}} dx \), we apply this formula using the given values.
The general formula for the Trapezoidal Rule is:
\[ T_{n} = \frac{b-a}{2n} \left[ f(a) + 2 \sum_{i=1}^{n-1} f(a+ih) + f(b) \right] \]
where \( a \) and \( b \) are the limits of integration, \( n \) is the number of trapezoids, and \( h = \frac{b-a}{n} \) is the uniform width of each trapezoid's base. The term \( f(a) \) represents the height at the beginning of the interval, while \( f(b) \) is the height at the end. The heights of the points between are accounted for and doubled since these heights are shared by two trapezoids. To approximate the integral \( \int_{0}^{2} e^{-x^{2}} dx \), we apply this formula using the given values.
Simpson's Rule
Moving to Simpson's Rule, we find a more sophisticated and generally more accurate method of numerical integration than the Trapezoidal Rule. Simpson's Rule uses parabolic arcs instead of straight lines to approximate the area under a curve, seeking to better match the curvature of the function.
The formula for Simpson’s Rule is as follows:
\[ S_{n} = \frac{b-a}{3n} \left[ f(a) + 4 \sum_{i=1}^{\frac{n}{2}} f(a + (2i-1)h) + 2 \sum_{i=1}^{\frac{n}{2} - 1} f(a + 2ih) + f(b) \right] \]
In this formula, \( a \), \( b \), \( n \), and \( h \) have the same meanings as in the Trapezoidal Rule. The coefficients 4 and 2 are used to weight the terms appropriately, as the method integrates over subintervals that cover two segments for each approximation. Therefore, applying Simpson's Rule to the integral \( \int_{0}^{2} e^{-x^{2}} dx \) with \( n=4 \) requires us to substitute the specified values and perform the calculation accordingly.
The formula for Simpson’s Rule is as follows:
\[ S_{n} = \frac{b-a}{3n} \left[ f(a) + 4 \sum_{i=1}^{\frac{n}{2}} f(a + (2i-1)h) + 2 \sum_{i=1}^{\frac{n}{2} - 1} f(a + 2ih) + f(b) \right] \]
In this formula, \( a \), \( b \), \( n \), and \( h \) have the same meanings as in the Trapezoidal Rule. The coefficients 4 and 2 are used to weight the terms appropriately, as the method integrates over subintervals that cover two segments for each approximation. Therefore, applying Simpson's Rule to the integral \( \int_{0}^{2} e^{-x^{2}} dx \) with \( n=4 \) requires us to substitute the specified values and perform the calculation accordingly.
Integral Approximation
Integral approximation is a fundamental topic within numerical analysis where we aim to estimate the value of a definite integral without determining its exact value. This is especially essential when dealing with integrals that are complex or impossible to solve analytically.
Techniques like the Trapezoidal Rule and Simpson's Rule come into play when we seek to approximate the area under a curve over a specified interval by using geometric shapes to 'fit' under the curve as closely as possible. These shapes, whether trapezoids or parabolic segments, create a sum that converges towards the actual integral value as the number of segments increases.
In practical applications, this allows us to calculate quantities like the distance traveled by an object when given its velocity function, or the total energy used over time from a varying power source, despite complex relationships between the variables.
Techniques like the Trapezoidal Rule and Simpson's Rule come into play when we seek to approximate the area under a curve over a specified interval by using geometric shapes to 'fit' under the curve as closely as possible. These shapes, whether trapezoids or parabolic segments, create a sum that converges towards the actual integral value as the number of segments increases.
In practical applications, this allows us to calculate quantities like the distance traveled by an object when given its velocity function, or the total energy used over time from a varying power source, despite complex relationships between the variables.
Numerical Analysis
Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis. It is an incredibly important field that intersects with all areas of engineering, science, and mathematics.
Numerical analysis involves a variety of methods to approximate solutions and includes error analysis to understand and control approximation errors. It not only encompasses methods for differential and integral equations but also covers techniques for solving equations, optimization, and performing numerical simulations.
Both the Trapezoidal Rule and Simpson's Rule are quintessential examples of numerical integration, a principal area in numerical analysis, demonstrating how we can find approximate solutions to integrals, which are a vital part of understanding and computing real-world problems where exact answers may not be feasible.
Numerical analysis involves a variety of methods to approximate solutions and includes error analysis to understand and control approximation errors. It not only encompasses methods for differential and integral equations but also covers techniques for solving equations, optimization, and performing numerical simulations.
Both the Trapezoidal Rule and Simpson's Rule are quintessential examples of numerical integration, a principal area in numerical analysis, demonstrating how we can find approximate solutions to integrals, which are a vital part of understanding and computing real-world problems where exact answers may not be feasible.
Other exercises in this chapter
Problem 21
Find the indefinite integral. (Hint: Integration by parts is not required for all the integrals.) $$ \int t \ln (t+1) d t $$
View solution Problem 22
Determine whether the improper integral diverges or converges. Evaluate the integral if it converges. $$ \int_{-\infty}^{\infty} x^{2} e^{-x^{3}} d x $$
View solution Problem 22
Use partial fractions to find the indefinite integral. $$ \int \frac{1}{4 x^{2}-9} d x $$
View solution Problem 22
Find the indefinite integral. (Hint: Integration by parts is not required for all the integrals.) $$ \int x^{3} \ln x d x $$
View solution