Chapter 12: Problem 44
Use a program similar to the Simpson's Rule program on page 906 to approximate the integral. Use \(n=100\). $$ \int_{1}^{4} x^{2} \sqrt{x+4} d x $$
Short Answer
Expert verified
The numerical approximation of the integral can be found by implementing each step in Simpson’s rule. Since this formula involves summations and a variable step size, determining the exact result requires evaluating the result of the Simpson's Rule formula with the given function and limits of integration.
Step by step solution
01
Formulate The Function
Firstly, define the function to be integrated. This function is \(f(x) = x^{2} \sqrt{x+4}\).
02
Define The Limits of Integration
The limits of the integration are noted from the integral symbol, with the lower limit being 1 and the upper limit being 4.
03
Decide The Number of Intervals
Next, determine the number of intervals. In this case, \(n = 100\), which means the function will be divided into 100 equal parts between x=1 and x=4.
04
Calculate the Width of Each Interval
Following this, calculate the width of each interval, \(h\), using the formula \(h = (b - a) / n\), where \(a\) and \(b\) are the lower and upper limits respectively. Here, \(h = (4-1) / 100 = 0.03\).
05
Implement Simpson's Rule
Simpson's rule is applied by the formula: \( \int_{a}^{b} f(x) \,dx \approx \frac{h}{3} [f(a) + 4\sum_{i=1}^{n/2} f(a+2i-1)h + 2\sum_{i=1}^{n/2-1} f(a+2ih) + f(b)]\). Add the starting and ending terms to the sums of the function evaluated at intermediate points for which their respective terms have weights of 2 and 4.
06
Computing the Value
Finally, compute the summation and substitute all necessary values into Simpson's Rule for the approximation.
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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 vital tool in mathematics used to approximate the value of an integral. Unlike analytical integration that provides an exact symbolic expression, numerical integration offers a practical way to calculate integrals that are difficult or impossible to solve analytically.
There are several methods available, such as the Trapezoidal Rule, Simpson's Rule, and Romberg Integration. Each method provides a different approach to estimate the area under a curve.
Simpson's Rule, for example, uses parabolas to approximate the segments between data points, which often results in a more accurate integral approximation when compared to other methods like the Trapezoidal Rule. By breaking an integral into smaller parts, numerical integration offers a powerful and accessible approach to problems involving curves that defy simple calculation.
With numerical integration, even complex functions can be handled effectively, providing solutions that are usable in a wide range of applications from engineering to physics.
There are several methods available, such as the Trapezoidal Rule, Simpson's Rule, and Romberg Integration. Each method provides a different approach to estimate the area under a curve.
Simpson's Rule, for example, uses parabolas to approximate the segments between data points, which often results in a more accurate integral approximation when compared to other methods like the Trapezoidal Rule. By breaking an integral into smaller parts, numerical integration offers a powerful and accessible approach to problems involving curves that defy simple calculation.
With numerical integration, even complex functions can be handled effectively, providing solutions that are usable in a wide range of applications from engineering to physics.
Integral Approximation
Integral approximation is the process of finding an approximate value for a definite integral. It is used extensively when exact solutions are difficult to obtain.
The accuracy of the approximation depends on the method used and the number of intervals (often denoted as \(n\)) in which the integral is divided.
Here are some key points:
The accuracy of the approximation depends on the method used and the number of intervals (often denoted as \(n\)) in which the integral is divided.
Here are some key points:
- More intervals generally lead to more accuracy.
- Differing methods, like Simpson's Rule or the Trapezoidal Rule, can offer varying precision levels.
- Choosing the correct method and interval count depends on the specific problem and required precision.
Calculus Concepts
Calculus concepts like integration play a significant role in mathematics and science, serving as the foundation for a variety of applications. Integral calculus focuses on finding the total accumulation of values, often represented as the area under a curve.
The specific example in the exercise involves the function \( f(x) = x^2 \sqrt{x+4} \), a combination of algebraic and transcedental elements. Solving it requires understanding:
By grasping these concepts, students can effectively analyze and interpret mathematical problems, transforming them into valuable solutions that serve various analytical purposes.
The specific example in the exercise involves the function \( f(x) = x^2 \sqrt{x+4} \), a combination of algebraic and transcedental elements. Solving it requires understanding:
- How to set limits of integration from the integral signs.
- The realization that some integrals can't be solved analytically, necessitating numerical methods.
By grasping these concepts, students can effectively analyze and interpret mathematical problems, transforming them into valuable solutions that serve various analytical purposes.
Advanced Mathematics
Advanced mathematics often encompasses complex computations and abstract concepts, building upon foundational knowledge in specific areas like calculus and algebra.
In the context of the original exercise, employing techniques such as Simpson's Rule highlights the need for a deeper understanding of numerical methods.
Essential skills include:
As students engage with these ideas, they develop the ability to think analytically and solve complex problems, preparing them for further exploration in mathematics.
In the context of the original exercise, employing techniques such as Simpson's Rule highlights the need for a deeper understanding of numerical methods.
Essential skills include:
- Recognizing when traditional methods can't provide solutions, leading to the use of numerical techniques.
- Applying formulas systematically to solve intricate problems.
- Interpreting results within the practical framework of real-world applications.
As students engage with these ideas, they develop the ability to think analytically and solve complex problems, preparing them for further exploration in mathematics.