Calculate the integral approximation Se for S6 = J [²4-²² d.z.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Calculate the integral approximation \( S_6 \) for \( \int_0^1 4 - x^2 \, dx \).**
\( S_6 = \)
[Input Box]
This problem asks you to calculate the numerical approximation \(S_6\) for the given definite integral \( \int_0^1 4 - x^2 \, dx \). To solve this, you typically use methods such as Simpson's rule, which involves finding areas under curves through polynomial approximations, specifically through parabolic segments.
1. \( \int_0^1 4 - x^2 \, dx \): This is the integral you need to approximate, it measures the area under the curve \( y = 4 - x^2 \) from \( x = 0 \) to \( x = 1 \).
2. \( S_6 \): This notation typically represents an approximation using Simpson's rule with \( n = 6 \) subintervals.
To solve this integral:
- Divide the interval [0,1] into 6 subintervals.
- Apply Simpson's rule formula:
\[ S_6 = \frac{\Delta x}{3} \left[f(x_0) + 4\sum_{i=1,3,5} f(x_i) + 2\sum_{i=2,4} f(x_i) + f(x_6) \right] \]
where \( \Delta x = \frac{1-0}{6} = \frac{1}{6} \), and \( x_i \) are the endpoints of the subintervals.
**Note**: Ensure that the calculations and plots, if there were any diagrams or graphical explanations involved, precisely follow the interval and function evaluations to achieve the correct approximation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F85a13651-0641-4e1f-9366-64e873b85445%2F3d1eacb6-53ae-47cb-8156-9c57e66b3d36%2Fst6yy67_processed.png&w=3840&q=75)
Transcribed Image Text:**Calculate the integral approximation \( S_6 \) for \( \int_0^1 4 - x^2 \, dx \).**
\( S_6 = \)
[Input Box]
This problem asks you to calculate the numerical approximation \(S_6\) for the given definite integral \( \int_0^1 4 - x^2 \, dx \). To solve this, you typically use methods such as Simpson's rule, which involves finding areas under curves through polynomial approximations, specifically through parabolic segments.
1. \( \int_0^1 4 - x^2 \, dx \): This is the integral you need to approximate, it measures the area under the curve \( y = 4 - x^2 \) from \( x = 0 \) to \( x = 1 \).
2. \( S_6 \): This notation typically represents an approximation using Simpson's rule with \( n = 6 \) subintervals.
To solve this integral:
- Divide the interval [0,1] into 6 subintervals.
- Apply Simpson's rule formula:
\[ S_6 = \frac{\Delta x}{3} \left[f(x_0) + 4\sum_{i=1,3,5} f(x_i) + 2\sum_{i=2,4} f(x_i) + f(x_6) \right] \]
where \( \Delta x = \frac{1-0}{6} = \frac{1}{6} \), and \( x_i \) are the endpoints of the subintervals.
**Note**: Ensure that the calculations and plots, if there were any diagrams or graphical explanations involved, precisely follow the interval and function evaluations to achieve the correct approximation.
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