Undergraduate
  1. Algebra  1.1. Linear Algebra  1.1.1. Matrices  1.1.2. Determinants  1.1.3. Systems of Linear Equations  1.1.4. Vector Spaces  1.1.5. Linear Transformations  1.1.6. Eigenvalues and Eigenvectors  1.1.7. Diagonalization  1.1.8. Inner Product Spaces  1.1.9. Orthogonality and Orthonormal Bases  1.1.10. Singular Value Decomposition  1.2. Abstract Algebra  1.2.1. Groups  1.2.2. Subgroups  1.2.3. Cyclic Groups  1.2.4. Permutation Groups  1.2.5. Homomorphisms  1.2.6. Normal Subgroups  1.2.7. Rings  1.2.8. Fields  1.2.9. Ideals and Quotient Rings  1.2.10. Polynomial Rings  1.2.11. Vector Spaces over Fields  1.2.12. Modules  1.2.13. Galois Theory
  2. Calculus  2.1. Differential Calculus  2.1.1. Limits  2.1.2. Continuity  2.1.3. Derivatives  2.1.4. Chain Rule  2.1.5. Implicit Differentiation  2.1.6. Applications of Derivatives  2.1.7. Optimization Problems  2.1.8. Mean Value Theorem  2.2. Integral Calculus  2.2.1. Definite Integrals  2.2.2. Indefinite Integrals  2.2.3. Techniques of Integration  2.2.4. Improper Integrals  2.2.5. Applications of Integration  2.2.6. Arc Length and Surface Area  2.3. Multivariable Calculus  2.3.1. Partial Derivatives  2.3.2. Gradient  2.3.3. Divergence and Curl  2.3.4. Multiple Integrals  2.3.5. Line Integrals  2.3.6. Surface Integrals  2.3.7. Green's Theorem  2.3.8. Stokes' Theorem  2.3.9. Divergence Theorem  2.3.10. Jacobians
  3. Differential Equations  3.1. Ordinary Differential Equations  3.1.1. First Order Differential Equations  3.1.2. Higher Order Differential Equations  3.1.3. Systems of Differential Equations  3.1.4. Series Solutions  3.1.5. Numerical Methods for ODEs  3.2. Partial Differential Equations  3.2.1. Heat Equation  3.2.2. Wave Equation  3.2.3. Laplace Equation  3.2.4. Separation of Variables  3.2.5. Fourier Transform Methods
  4. Real Analysis  4.1. Sequences and Series  4.1.1. Convergence of Sequences  4.1.2. Series Tests  4.1.3. Power Series  4.1.4. Uniform Convergence  4.2. Functions of Real Variables  4.2.1. Limits and Continuity  4.2.2. Uniform Continuity  4.2.3. Differentiation in Functions of Real Variables  4.2.4. Riemann Integration  4.2.5. Lebesgue Integration
  5. Complex Analysis  5.1. Complex Numbers  5.1.1. Polar Form  5.1.2. Exponential Form  5.2. Functions of a Complex Variable  5.2.1. Analytic Functions  5.2.2. Cauchy-Riemann Equations  5.2.3. Contour Integrals  5.2.4. Cauchy's Integral Theorem  5.2.5. Residue Theorem  5.2.6. Conformal Mapping
  6. Probability and Statistics  6.1. Probability Theory  6.1.1. Basic Probability Concepts  6.1.2. Random Variables  6.1.3. Probability Distributions  6.1.4. Expectation and Variance  6.1.5. Conditional Probability and Bayes' Theorem  6.2. Statistics  6.2.1. Descriptive Statistics  6.2.2. Inferential Statistics  6.2.3. Hypothesis Testing  6.2.4. Regression Analysis  6.2.5. ANOVA
  7. Numerical Methods  7.1. Root-Finding Algorithms  7.2. Numerical Integration  7.3. Numerical Differentiation  7.4. Numerical Solutions to Differential Equations  7.5. Matrix Computations  7.6. Interpolation and Extrapolation
  8. Discrete Mathematics  8.1. Logic and Proof  8.2. Combinatorics  8.3. Graph Theory  8.4. Boolean Algebra  8.5. Recurrence Relations
  9. Linear Programming and Optimization  9.1. Simplex Method  9.2. Duality  9.3. Integer Programming  9.4. Nonlinear Optimization
  10. Topology  10.1. Metric Spaces  10.2. Topological Spaces  10.3. Continuity and Homeomorphisms  10.4. Compactness  10.5. Connectedness
  11. Geometry  11.1. Euclidean Geometry  11.2. Differential Geometry  11.3. Non-Euclidean Geometry  11.4. Projective Geometry
  12. Mathematical Physics  12.1. Fourier Series  12.2. Laplace Transforms  12.3. Applications of Differential Equations  12.4. Special Functions
  13. Set Theory and Logic  13.1. Sets and Subsets  13.2. Relations and Functions  13.3. Cardinality  13.4. Formal Logic  13.5. Zermelo-Fraenkel Set Theory
  14. Functional Analysis  14.1. Normed Spaces  14.2. Banach Spaces  14.3. Hilbert Spaces  14.4. Linear Operators

Undergraduate → Numerical Methods ↓


Numerical Integration


Numerical integration is a method used to estimate the value of integrals. Integration is one of the two fundamental operations in calculus, along with differentiation. While some integrals can be solved analytically, meaning we can find a closed-form expression for the integral, others are more complicated and require approximation methods. Numerical integration deals with the latter scenario. This topic is particularly relevant when dealing with functions for which it is difficult or impossible to obtain an analytical integral.

What is integration?

Before diving into numerical integration, let's briefly recall what integration means in the context of calculus. Integration is the process of finding the integral of a function. In simple terms, if differentiation is about finding the rate of change or slope of a function, integration is about finding the total value, accumulation, or area under the curve of that function.

Mathematically, the definite integral of a function f(x) over the interval [a, b] is represented as:

∫ a b f(x) dx

The solution of this integral gives us the area enclosed between the function f(x), the x-axis and the vertical lines at x = a and x = b.

When is numerical integration used?

Numerical integration is used when:

  • This problem is so complex that finding an exact analytical solution is challenging.
  • The data is only available at discrete points, not as a continuous function.
  • An estimate is often sufficient for some task, such as in engineering applications or simulations.

Original idea

The main idea behind numerical integration is to approximate the area under a curve by dividing it into simpler geometric shapes whose areas we can easily calculate. The simplest method begins by using rectangles, which we will discuss shortly.

Rectangular approximation methods

Let's look at the simplest way to estimate the integral: rectangular estimation. Such methods involve dividing the area under the curve into several rectangles and then adding their areas.

Left Riemann equation

Left Riemann summation is a method in which we approximate integrals using rectangles. The height of each rectangle is determined by the value of the function at the left endpoint of each subinterval. Essentially, if we want to approximate ∫ a b f(x) dx, we divide [a, b] into n subintervals of equal width Δx, which is calculated as:

Δx = (b – a) / n

In a left Riemann sum, the region A is approximated by the sum:

A ≈ Σ f(x i)Δx, from i = 0 to n-1

Right Riemann sum

The right Riemann sum is similar to the left Riemann sum, but uses the right endpoint of each subinterval to determine the heights of the rectangles:

A ≈ Σ f(x i+1)Δx, from i = 0 to n-1

Midpoint rule

In the midpoint rule, we use the midpoint of each subinterval to determine the height of the rectangle. The estimate is:

A ≈ Σ f((x i + x i+1) / 2)Δx, from i = 0 to n-1

Trapezoidal rule

The trapezoidal rule improves upon rectangular methods by fitting a trapezoid under each segment of the curve. This is done effectively by taking the average of the left and right endpoints of the subintervals:

A ≈ (Δx/2) [f(x 0) + 2f(x 1) + 2f(x 2) + ... + 2f(x n-1) + f(x n)]

Simpson's rule

Simpson's rule goes a step further by fitting parabolas to each pair of subintervals instead of straight lines. This gives significantly better estimates than both the rectangular methods and the trapezoidal rule for smooth functions:

A ≈ (Δx/3) [f(x 0) + 4f(x 1) + 2f(x 2) + ... + 4f(x n-1) + f(x n)]

Handling more complex tasks and data

For functions with discontinuities, rapidly changing slopes, or limited data points, advanced techniques are necessary. Adaptive integration methods dynamically adjust sub-intervals based on the rate of function changes. These methods are more sophisticated but offer better accuracy and performance. Popular adaptive methods include adaptive quadrature and Monte Carlo integration for functions with greater degrees of complexity.

Common challenges in numerical integration

Despite the computational efficiency of numerical methods, several challenges demand attention:

  • Rounding error: The accumulation of small errors over many calculations can result in significant inaccuracy.
  • Precision: Deciding on the appropriate number of subintervals or the appropriate tolerance level can determine the precision of the results.
  • Working behavior: Uneven or steep gradients, discontinuities or oscillations may reduce accuracy.

Practical applications of numerical integration

Numerical integration is widely used in many fields, including:

  • Engineering: Computing in area, volume, displacement, and signal processing.
  • Physics: Calculating work done and other integral-based measurements.
  • Statistics: Integrating probability density functions to determine cumulative distribution functions.
  • Economics: Calculation of consumer and producer surplus and welfare analysis.

Conclusion

Numerical integration provides powerful techniques for approximating complex integrals, which plays an essential role in mathematics, engineering, physics, and beyond. Understanding how various methods are derived, implemented, and applied helps us solve computational problems efficiently. Continuous refinements in numerical methods, computational power, and algorithms increase the possibilities for solving complex mathematical challenges with accuracy.


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