Answer to Question #260848 in English for Zee

Question

Simple trapezoidal rule for integration.

Determine the Lagrange interpolation polynomial through the points (-1;f(-1)) and (1;f(1)) and use it to derive the simple trapezoidal rule for integration over the interval [-1;1]

Answer 1

Trapezoidal Rule is a rule that evaluates the area under the curves by dividing the total area into smaller trapezoids rather than using rectangles. This integration works by approximating the region under the graph of a function as a trapezoid, and it calculates the area.T n = 1 2 Δ x ( f ( x 0 ) + 2 f ( x 1 ) + 2 f ( x 2 ) + ⋯ + 2 f ( x n − 1 ) + f ( x n ) ) . Then, lim n → + ∞ T n = ∫ a b f ( x ) d x