Numerical Integration

Numerical integration, also known as quadrature, is a collection of techniques used to approximate the definite integral of a function. This is essential when:

• Analytical solutions are intractable: Finding the antiderivative of a function can be extremely difficult or impossible.

• The function is only known at specific points: Data might be collected experimentally or through simulations.

Key Concepts: - Approximation: Numerical integration methods approximate the area under the curve of a function by dividing it into smaller shapes (rectangles, trapezoids, etc.) and summing their areas. - Accuracy: The accuracy of the approximation depends on the method used, the number of subdivisions, and the smoothness of the function.

Common Methods: - Riemann Sums: Basic approach using rectangles. - Trapezoidal Rule: Approximates the curve with straight lines. - Simpson’s Rule: Approximates the curve with parabolic arcs. - Gaussian Quadrature: Uses strategically placed points for higher accuracy.

Applications:

• Engineering: Calculating work done by a variable force, determining the volume of irregular shapes.

• Physics: Finding the center of mass, calculating probabilities in statistical mechanics.

• Finance: Pricing options and other financial derivatives.

• Computer Graphics: Rendering realistic images and animations.