Chapter 2: Problem 32
Approximate the integral to three decimal places using the indicated rule. \(\int_{0}^{0.8} e^{-x^{2}} d x ;\) Simpson's rule; \(n=4\)
Short Answer
Expert verified
The approximate value is 0.658.
Step by step solution
01
Understanding Simpson's Rule
Simpson's Rule is a method for numerical integration that estimates the integral of a function using parabolic arcs instead of straight-line segments. The formula for Simpson's Rule is given by: \[\int_{a}^{b} f(x) \, dx \approx \frac{h}{3} \left[f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + \, ... \, + 2f(x_{n-2}) + 4f(x_{n-1}) + f(x_n) \right]\]where \( n \) must be even and \( h = \frac{b-a}{n} \).
02
Define Parameters
For this problem, we have the integral \( \int_0^{0.8} e^{-x^2} \, dx \). Here, \( a = 0 \), \( b = 0.8 \), and \( n = 4 \). Calculate \( h \): \[h = \frac{0.8-0}{4} = 0.2\]This means the subinterval width is 0.2.
03
Calculate Function Values
Determine the values of the function \( f(x) = e^{-x^2} \) at each point:- \( f(x_0) = e^{-(0)^2} = e^0 = 1 \)- \( f(x_1) = e^{-(0.2)^2} \approx e^{-0.04} \approx 0.9608 \)- \( f(x_2) = e^{-(0.4)^2} \approx e^{-0.16} \approx 0.8521 \)- \( f(x_3) = e^{-(0.6)^2} \approx e^{-0.36} \approx 0.6977 \)- \( f(x_4) = e^{-(0.8)^2} \approx e^{-0.64} \approx 0.5273 \) These are needed in the Simpson's Rule formula.
04
Apply Simpson's Rule Formula
Substitute the calculated function values into the Simpson's Rule formula:\[\int_{0}^{0.8} e^{-x^2} dx \approx \frac{0.2}{3} \left[1 + 4(0.9608) + 2(0.8521) + 4(0.6977) + 0.5273 \right]\]Simplify the expression:\[\int_{0}^{0.8} e^{-x^2} dx \approx \frac{0.2}{3} \left[1 + 3.8432 + 1.7042 + 2.7908 + 0.5273 \right]\]Calculate the inner sum:\[ = 1 + 3.8432 + 1.7042 + 2.7908 + 0.5273 = 9.8655\]Compute the expression:\[\approx \frac{0.2}{3} \times 9.8655 \approx \frac{1.9731}{3} \approx 0.6577\]
05
Round the Result
Round the final value to three decimal places to obtain the approximation:\[\int_{0}^{0.8} e^{-x^2} dx \approx 0.658\]This is the approximation of the integral using Simpson's rule with four subintervals.
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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 an essential technique in calculus and applied mathematics used to approximate the value of definite integrals. When a function is complex, or its antiderivative is difficult or impossible to determine analytically, numerical integration offers an effective solution.
By using numerical methods, we can approximate the area under the curve represented by a function over a given interval. This is particularly useful in fields like engineering, physics, and computer science, where precise data about seemingly non-integrable functions is necessary.
The process involves discretizing the continuous interval of the integral into smaller subintervals and approximating the integral using specific mathematical rules. These rules, such as the Trapezoidal Rule, Simpson’s Rule, and more, help calculate approximate values with varying degrees of accuracy. Each method has its own advantages and applications depending on the nature of the function and the desired accuracy.
By using numerical methods, we can approximate the area under the curve represented by a function over a given interval. This is particularly useful in fields like engineering, physics, and computer science, where precise data about seemingly non-integrable functions is necessary.
The process involves discretizing the continuous interval of the integral into smaller subintervals and approximating the integral using specific mathematical rules. These rules, such as the Trapezoidal Rule, Simpson’s Rule, and more, help calculate approximate values with varying degrees of accuracy. Each method has its own advantages and applications depending on the nature of the function and the desired accuracy.
Integral Approximation
Integral approximation is a critical component of numerical integration. It involves estimating the value of a definite integral, especially when direct calculation is infeasible. For many real-world applications, relying on approximations of integrals allows for efficient and practical computations.
To approximate integrals, we often use methods like Simpson's Rule, where the integral of the function is approximated using parabolic segments. Such methods rely on sampling a function at specific intervals and then combining these samples in a weighted manner to approximate the total area under the curve.
With integral approximation, the focus is not only on finding the value but also on minimizing errors associated with the approximation. Hence, dividing the interval into smaller subintervals or using higher-order functions to approximate the curve often leads to more accurate results. It’s a balance of computational efficiency and precision.
To approximate integrals, we often use methods like Simpson's Rule, where the integral of the function is approximated using parabolic segments. Such methods rely on sampling a function at specific intervals and then combining these samples in a weighted manner to approximate the total area under the curve.
With integral approximation, the focus is not only on finding the value but also on minimizing errors associated with the approximation. Hence, dividing the interval into smaller subintervals or using higher-order functions to approximate the curve often leads to more accurate results. It’s a balance of computational efficiency and precision.
Mathematical Functions
Understanding mathematical functions is crucial when dealing with numerical integration and integral approximation. A function essentially maps an input to an output, forming the basis for analyzing and solving complex mathematical problems.
When working on integration, particularly over a specific interval \( [a, b] \), it's important to comprehend the behavior of the function. In the context of our example, the function \( f(x) = e^{-x^2} \) represents a continuous curve that needs to be integrated between 0 and 0.8.
Evaluating how a function behaves at various points is crucial for applying numerical rules like Simpson’s. The values of the function at specified intervals are used to estimate the area under the curve, and these discrete points help form an approximate integral representation when exact calculation methods are not applicable. Such understanding and manipulation of mathematical functions are foundational skills in both academic problems and practical applications.
When working on integration, particularly over a specific interval \( [a, b] \), it's important to comprehend the behavior of the function. In the context of our example, the function \( f(x) = e^{-x^2} \) represents a continuous curve that needs to be integrated between 0 and 0.8.
Evaluating how a function behaves at various points is crucial for applying numerical rules like Simpson’s. The values of the function at specified intervals are used to estimate the area under the curve, and these discrete points help form an approximate integral representation when exact calculation methods are not applicable. Such understanding and manipulation of mathematical functions are foundational skills in both academic problems and practical applications.
