Simpson's 3/8 Rule is another high-accuracy numerical integration formula derived by fitting cubic polynomials across groups of four consecutive points over subintervals.
It is particularly useful when the total number of subintervals n n n multiple of 3 (e.g. n = 3 , 6 , 9 , 12 n = 3, 6, 9, 12 n = 3 , 6 , 9 , 12
Formula for Simpson's 3/8 Rule:
I = ∫ a b f ( x ) , d x ≈ 3 h 8 [ ( y 0 + y n ) + 3 ( y 1 + y 2 + y 4 + y 5 + … ) + 2 ( y 3 + y 6 + … ) ] I = \int_{a}^{b} f(x) , dx \approx \frac{3h}{8} \left[ (y_0 + y_n) + 3(y_1 + y_2 + y_4 + y_5 + \dots) + 2(y_3 + y_6 + \dots) \right] I = ∫ a b f ( x ) , d x ≈ 8 3 h [ ( y 0 + y n ) + 3 ( y 1 + y 2 + y 4 + y 5 + … ) + 2 ( y 3 + y 6 + … ) ] Where:
a a a b b b n n n must be a multiple of 3 ).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 2 + y 4 + d o t s y_1 + y_2 + y_4 + \\dots y 1 + y 2 + y 4 + d o t s y 3 + y 6 + d o t s y_3 + y_6 + \\dots y 3 + y 6 + d o t s Example of Simpson's 3/8 Rule Evaluate Definite Integral
∫ 0 3 e x , d x \int_{0}^{3} e^x , dx ∫ 0 3 e x , d x With n = 3 subintervals (multiple of 3) n = 3 \text{ subintervals (multiple of 3)} n = 3 subintervals (multiple of 3)
Step 1: Find Step Size (h h h h = 3 − 0 3 = 1.0 h = \frac{3 - 0}{3} = 1.0 h = 3 3 − 0 = 1.0 Step 2: Determine Grid Points (x i x_i x i x 0 = 0.0 , x 1 = 1.0 , x 2 = 2.0 , x 3 = 3.0 x_0 = 0.0, \quad x_1 = 1.0, \quad x_2 = 2.0, \quad x_3 = 3.0 x 0 = 0.0 , x 1 = 1.0 , x 2 = 2.0 , x 3 = 3.0 Step 3: Evaluate Function Values (y i = e x i y_i = e^{x_i} y i = e x i y 0 = 1.00000 , y 1 = 2.71828 y_0 = 1.00000, \quad y_1 = 2.71828 y 0 = 1.00000 , y 1 = 2.71828 y 2 = 7.38906 , y 3 = 20.08554 y_2 = 7.38906, \quad y_3 = 20.08554 y 2 = 7.38906 , y 3 = 20.08554 Step 4 & 5: Apply Simpson's 3/8 Formula I ≈ 3 ( 1.0 ) 8 [ ( 1.0 + 20.08554 ) + 3 ( 2.71828 + 7.38906 ) ] I \approx \frac{3(1.0)}{8} \left[ (1.0 + 20.08554) + 3(2.71828 + 7.38906) \right] I ≈ 8 3 ( 1.0 ) [ ( 1.0 + 20.08554 ) + 3 ( 2.71828 + 7.38906 ) ] I ≈ 0.375 × 51.40756 ≈ 19.2778 I \approx 0.375 \times 51.40756 \approx 19.2778 I ≈ 0.375 × 51.40756 ≈ 19.2778 Conclusion ∫ 0 3 e x , d x ≈ 19.2778 \boxed{\int_{0}^{3} e^x , dx \approx 19.2778} ∫ 0 3 e x , d x ≈ 19.2778 Therefore, using Simpson's 3/8 Rule with n = 3 n = 3 n = 3 19.2778 (exact value e 3 − 1 ≈ 19.0855 e^3 - 1 \approx 19.0855 e 3 − 1 ≈ 19.0855
Interactive Simpson's 3/8 Rule Calculator Simpson's 3/8 Rule Solver Demo
