Simpson's 1/3 Rule is an advanced numerical integration technique that approximates the definite integral ∫ a b f ( x ) d x \int_{a}^{b} f(x) \, dx ∫ a b f ( x ) d x
Because it uses 2nd-degree polynomials, it provides significantly higher accuracy than the Trapezoidal Rule. A key requirement of Simpson's 1/3 Rule is that the number of subintervals n n n must be even .
Formula for Simpson's 1/3 Rule:
I = ∫ a b f ( x ) , d x ≈ h 3 [ ( y 0 + y n ) + 4 ( y 1 + y 3 + … ) + 2 ( y 2 + y 4 + … ) ] I = \int_{a}^{b} f(x) , dx \approx \frac{h}{3} \left[ (y_0 + y_n) + 4(y_1 + y_3 + \dots) + 2(y_2 + y_4 + \dots) \right] I = ∫ a b f ( x ) , d x ≈ 3 h [ ( y 0 + y n ) + 4 ( y 1 + y 3 + … ) + 2 ( y 2 + y 4 + … ) ] Where:
a a a b b b n n n must be an even integer ).h = b − a n h = \frac{b - a}{n} h = n b − a y 0 + y n y_0 + y_n y 0 + y n y 1 + y 3 + d o t s y_1 + y_3 + \\dots y 1 + y 3 + d o t s y 2 + y 4 + d o t s y_2 + y_4 + \\dots y 2 + y 4 + d o t s Example of Simpson's 1/3 Rule Evaluate Definite Integral
∫ 0 1 1 1 + x , d x \int_{0}^{1} \frac{1}{1 + x} , dx ∫ 0 1 1 + x 1 , d x With n = 4 subintervals (even) n = 4 \text{ subintervals (even)} n = 4 subintervals (even)
Step 1: Find Step Size (h h h h = 1 − 0 4 = 0.25 h = \frac{1 - 0}{4} = 0.25 h = 4 1 − 0 = 0.25 Step 2: Determine Grid Points (x i x_i x i x 0 = 0.0 , x 1 = 0.25 , x 2 = 0.5 x_0 = 0.0, \quad x_1 = 0.25, \quad x_2 = 0.5 x 0 = 0.0 , x 1 = 0.25 , x 2 = 0.5 x 3 = 0.75 , x 4 = 1.0 x_3 = 0.75, \quad x_4 = 1.0 x 3 = 0.75 , x 4 = 1.0 Step 3: Evaluate Function Values (y i = 1 1 + x i y_i = \frac{1}{1 + x_i} y i = 1 + x i 1 y 0 = 1.0 , y 1 = 0.8 , y 2 = 0.6667 y_0 = 1.0, \quad y_1 = 0.8, \quad y_2 = 0.6667 y 0 = 1.0 , y 1 = 0.8 , y 2 = 0.6667 y 3 = 0.5714 , y 4 = 0.5 y_3 = 0.5714, \quad y_4 = 0.5 y 3 = 0.5714 , y 4 = 0.5 Step 4 & 5: Apply Simpson's 1/3 Formula I ≈ 0.25 3 [ ( 1.0 + 0.5 ) + 4 ( 0.8 + 0.5714 ) + 2 ( 0.6667 ) ] I \approx \frac{0.25}{3} \left[ (1.0 + 0.5) + 4(0.8 + 0.5714) + 2(0.6667) \right] I ≈ 3 0.25 [ ( 1.0 + 0.5 ) + 4 ( 0.8 + 0.5714 ) + 2 ( 0.6667 ) ] I ≈ 0.25 3 [ 8.31906 ] ≈ 0.69315 I \approx \frac{0.25}{3} \left[ 8.31906 \right] \approx 0.69315 I ≈ 3 0.25 [ 8.31906 ] ≈ 0.69315 Conclusion ∫ 0 1 1 1 + x , d x ≈ 0.69315 \boxed{\int_{0}^{1} \frac{1}{1 + x} , dx \approx 0.69315} ∫ 0 1 1 + x 1 , d x ≈ 0.69315 Therefore, using Simpson's 1/3 Rule with n = 4 n = 4 n = 4 0.69315 (exact value ln 2 ≈ 0.69315 \ln 2 \approx 0.69315 ln 2 ≈ 0.69315
Interactive Simpson's 1/3 Rule Calculator Simpson's 1/3 Rule Solver Demo
