The formula to calculate the approximation of a definite integral using Simpson's 1/3 Rule is:
\[ I = \frac{h}{3} (y_0 + 4y_1 + y_2) \]
Where:
Simpson's 1/3 Rule is a numerical integration technique used to approximate definite integrals. It works by dividing the area under a curve into a series of parabolic arcs, then summing the areas of these arcs to estimate the total area. The rule is named "1/3" because each segment's area is calculated as \( \frac{h}{3} (y_0 + 4y_1 + y_2) \), where \( h \) is the width of the segment and \( y_0 \), \( y_1 \), and \( y_2 \) are the function values at the left, middle, and right endpoints of the segment, respectively. This method is more accurate than the Trapezoidal Rule for functions that are not linear.
Let's assume the following values:
Step 1: Calculate the Approximation of the Definite Integral (I):
\[ I = \frac{1}{3} (1 + 4 \times 4 + 1) = \frac{1}{3} (1 + 16 + 1) = \frac{1}{3} \times 18 = 6 \]
The approximation of the definite integral is 6.