Numerical Integration
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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.