Newton-Cotes Integration Formulas

• Covers numerical integration techniques

Newton-Cotes Integration Formulas

• These formulas approximate a complicated or hard-to-integrate function with a simpler approximating function.

• A function is replaced with an approximating function that is easy to integrate such as a polynomial, $f_n(x)$.

• The integral $I$ from $a$ to $b$ of $f(x)$ is approximately equal to the integral from $a$ to $b$ of $f_n(x)$.

  • $I = \int_a^b f(x) dx \cong \int_a^b f_n(x) dx$ $$\int_a^b f_n(x) dx = \int_a^b (a_0 + a_1 x + a_2 x^2 +... + a_n x^n) dx$$

Trapezoidal Rule

• Approximates the function as a straight line ($n = 1$).
  • $I = (b - a) \frac{f(a) + f(b)}{2}$
• The error, $E_t$, is calculated as $- \frac{1}{12} f''(\xi)(b - a)^3$

Simpson's 1/3 Rule

• Approximates the function as a parabola ($n = 2$).
  • $I = \frac{h}{3} [f(x_0) + 4f(x_1) + f(x_2)]$
  • $h = \frac{b - a}{2}$, $x_0 = a$, $x_1 = \frac{a + b}{2}$, $x_2 = b$
• The error, $E_t$, is calculated as $- \frac{1}{90} f^{(4)}(\xi)h^5$

Simpson's 3/8 Rule

• Approximates the function with a 3rd order polynomial ($n = 3$).
  • $I = \frac{3h}{8} [f(x_0) + 3f(x_1) + 3f(x_2) + f(x_3)]$
  • $h = \frac{b - a}{3}$, $x_0 = a$, $x_1 = a + h$, $x_2 = a + 2h$, $x_3 = b$
• The error, $E_t$, is calculated as $- \frac{3}{80} f^{(4)}(\xi)h^5$

Multiple Application of Trapezoidal Rule

• Divides the integration interval into multiple segments and applies the Trapezoidal rule to each.
  • $\int_a^b f(x) dx = \int_{x_0}^{x_1} f(x) dx + \int_{x_1}^{x_2} f(x) dx +... + \int_{x_{n-1}}^{x_n} f(x) dx$
  • $\int_{x_i}^{x_{i+1}} f(x) dx = (x_{i+1} - x_i) \frac{f(x_i) + f(x_{i+1})}{2}$
  • $I = (b - a) \frac{f(x_0) + 2 \sum_{i=1}^{n-1} f(x_i) + f(x_n)}{2n}$, where $h = \frac{b - a}{n}$
• The error $E$ is $- \frac{(b - a)^3}{12n^2} \overline{f''}$

Multiple Application of Simpson's 1/3 Rule

• Divides the integration interval into multiple segments and applies Simpson's 1/3 rule to each; the number of segments $n$ must be even.
  • $\int_a^b f(x) dx = \int_{x_0}^{x_2} f(x) dx + \int_{x_2}^{x_4} f(x) dx +... + \int_{x_{n-2}}^{x_n} f(x) dx$
  • $I = \frac{h}{3} [f(x_0) + 4 \sum_{i=1, 3, 5}^{n-1} f(x_i) + 2 \sum_{i=2, 4, 6}^{n-2} f(x_i) + f(x_n)]$, where $h = \frac{b - a}{n}$
• The error $E_a$ is $- \frac{(b - a)^5}{180 n^4} \overline{f^{(4)}}$

Example Calculation

• Two-segment Trapezoidal rule is used to estimate the integral of $f(x) = 0.2 + 25x - 200x^2 + 675x^3 - 900x^4 + 400x^5$ from $a = 0$ to $b = 0.8$.
• The analytical solution is 1.640533
• With $h = \frac{0.8 - 0}{2} = 0.4$, $I = (0.8 - 0) \frac{f(0) + 2f(0.4) + f(0.8)}{2 \times 2}$
• Given $f(0) = 0.2$, $f(0.4) = 2.456$, and $f(0.8) = -0.232$, the calculated integral $I$ is 0.8672.
• The error $E_t$ is calculated as 47.14%.