Simpson’s 1/3 Rule Calculator

Simpson's 1/3 Rule is a method for numerical integration, used to estimate the definite integral of a function. It is particularly useful for approximating the area under a curve when the function is difficult to integrate analytically.

Formula

The Simpson's 1/3 Rule formula for approximating the integral is:

\[ \int{a}^{b} f(x) dx \approx \frac{h}{3} \left[ f(a) + 4 \sum{i=1,3,5,\dots}^{n-1} f(xi) + 2 \sum{i=2,4,6,\dots}^{n-2} f(x_i) + f(b) \right] \]

Where:

• \( h = \frac{b-a}{n} \) is the width of each interval.
• \( n \) is the number of intervals (must be even).

Example Calculation

To approximate \(\int_{0}^{2} x^2 dx\) using Simpson's 1/3 Rule with \( n = 2 \):

• Function: \( f(x) = x^2 \)
• Limits: \( a = 0 \), \( b = 2 \)
• Interval width: \( h = \frac{2-0}{2} = 1 \)

Substituting these values into the formula, the result will be calculated accordingly.