4.3/

Numerical Quadrature

1. What?
2. How?
3. Based on?:
4. Method:
5. The Quadrature Formula:
   
6. The Error:
   

The Trapezoidal Rule

1. What?
2. The Trapezoidal Rule:
   
   • Derivation:
3. Error:
   

Simpson’s Rule

1. What?
   
2. Simpson’s Rule:
   
   • Derivation:
3. Error:
   

Measuring Precision

1. What?
   
2. Precision [degree of accuracy]:
   
3. Precision of Quadrature Formulas:
   
   • The degree of precision of a quadrature formula is
   • The Trapezoidal and Simpson’s rules are examples of \(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\)
   • Types of Newton-Cotes formulas:

Closed Newton-Cotes Formulas

1. What?
   
   • It is called closed because:
2. Form of the Formula:
   

   where,
   

3. The Error Analysis:
   
4. Degree of Preceision:
   
   • Even-n: the degree of precision is \(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\)
   • Odd-n: the degree of precision is
5. Closed Form Formulas:
   
   • \(n = 1\): \(\ \ \ \ \ \ \ \ \ \ \ \ \ \\) rule
   • \(n = 2\): \(\ \ \ \ \ \ \ \ \ \ \ \ \ \\) rule
   • \(n = 3\): \(\ \ \ \ \ \ \ \ \ \ \ \ \ \\) rule
   • n = 4:

Open Newton-Cotes Formulas

1. What?
   
   • They \(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\)
   • They use
   • This implies that
   • Open formulas contain \(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\)
2. Form of the Formula:
   

   where,
   

3. The Error Analysis:
   
4. Degree of Preceision:
   
   • Even-n:
   • Odd-n:
5. Open Form Formulas:
   
   • \(n = 0\): [PUT NAME HERE]
   • \(n = 1\):
   • \(n = 2\):
   • n = 3: