Q 21. Show that the summation expressi... [FREE SOLUTION]

Chapter 5: Q 21. (page 491)

Show that the summation expression for Simpson鈥檚 Rule in Theorem 5.29 follows from repeated applications of the formula for the definite integral of a quadratic in Theorem 5.28, in the case where n = 4 (i.e., where there are $\frac{n}{2} = 2$ parabolas).

Short Answer

Expert verified

Answer is $\begin{array}{rcl} {鈭�}_{x_{0}}^{x_{4}} p (x) d x & = & \frac{螖 x}{3} (p (x_{0}) + 4 p (x_{1}) + 2 p (x_{2}) + 4 p (x_{3}) + p (x_{4})) \end{array}$

Step by step solution

Step 1. Given information

Value of n is 4.

Step 2. Explanation

$\begin{array}{rcl} {鈭�}_{x_{0}}^{x_{2}} p (x) d x + {鈭�}_{x_{2}}^{x_{4}} p (x) d x & = & \frac{螖 x}{3} (p (x_{0}) + 4 p (x_{1}) + p (x_{2})) + \frac{螖 x}{3} (p (x_{2}) + 4 p (x_{3}) + p (x_{4})) \\ {鈭�}_{x_{0}}^{x_{4}} p (x) d x & = & \frac{螖 x}{3} (p (x_{0}) + 4 p (x_{1}) + p (x_{2}) + p (x_{2}) + 4 p (x_{3}) + p (x_{4})) \\ {鈭�}_{x_{0}}^{x_{4}} p (x) d x & = & \frac{螖 x}{3} (p (x_{0}) + 4 p (x_{1}) + 2 p (x_{2}) + 4 p (x_{3}) + p (x_{4})) \end{array}$

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Show that the summation expression for Simpson鈥檚 Rule in Theorem 5.29 follows from repeated applications of the formula for the definite integral of a quadratic in Theorem 5.28, in the case where n = 4 (i.e., where there are $\frac{n}{2} = 2$ parabolas).

Short Answer

Step by step solution

Step 1. Given information

Step 2. Explanation

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Show that the summation expression for Simpson鈥檚 Rule in Theorem 5.29 follows from repeated applications of the formula for the definite integral of a quadratic in Theorem 5.28, in the case where n = 4 (i.e., where there are n2=2parabolas).

Short Answer

Step by step solution

Step 1. Given information

Step 2. Explanation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Applied Mathematics

Theoretical and Mathematical Physics

Discrete Mathematics

Mechanics Maths

Decision Maths

Study anywhere. Anytime. Across all devices.

Show that the summation expression for Simpson鈥檚 Rule in Theorem 5.29 follows from repeated applications of the formula for the definite integral of a quadratic in Theorem 5.28, in the case where n = 4 (i.e., where there are $\frac{n}{2} = 2$ parabolas).