Composite Trapezoidal Rule Calculator

Formula

To calculate the approximation of the integral using the Composite Trapezoidal Rule:

\[ I = \frac{h}{2} \left( y_0 + 2y_1 + 2y_2 + \ldots + 2y_{n-1} + y_n \right) \]

Where:

\(I\) is the approximation of the integral
\(h\) is the width of the subintervals
\(y_0, y_1, \ldots, y_n\) are the function values at the nodes

What is the Composite Trapezoidal Rule?

The Composite Trapezoidal Rule is a numerical integration method used to approximate the definite integral of a function. It works by dividing the area under the curve of a function into several trapezoids, calculating the area of each, and then summing these areas to provide an approximation of the total area, and hence, the integral. The accuracy of the approximation increases with the number of trapezoids used.

Example Calculation

Let's assume the following values:

Width of Subintervals (\(h\)) = 0.5
Function Values (\(y_0, y_1, y_2, y_3\)) = 1, 2, 1.5, 1

Using the formula:

\[ I = \frac{0.5}{2} \left( 1 + 2 \cdot 2 + 2 \cdot 1.5 + 1 \right) = 0.25 \left( 1 + 4 + 3 + 1 \right) = 0.25 \cdot 9 = 2.25 \]

The approximation of the integral is 2.25.