🏃🏽‍♀️‍➡️Intro to Mathematical Analysis Unit 9 – The Riemann Integral

Definition and Concept

• The Riemann integral provides a precise definition of the integral of a function on a closed interval $[a, b]$
• Partitions the interval $[a, b]$ into subintervals and approximates the area under the curve using rectangles
• Considers the limit of the sum of these rectangular areas as the width of the subintervals approaches zero
• Denoted as $\int_a^b f(x) dx$, where $f(x)$ is the integrand and $[a, b]$ is the interval of integration
• The Riemann integral is a fundamental concept in calculus and mathematical analysis
  • Allows for the calculation of areas, volumes, and other quantities
  • Serves as a basis for more advanced integration techniques (Lebesgue integration)

Historical Context

• Developed by German mathematician Bernhard Riemann in the 19th century
• Riemann's work built upon the ideas of Cauchy and Darboux regarding the definition of the integral
• Riemann's definition provided a more rigorous and general approach to integration compared to earlier methods
  • Earlier methods relied on geometric intuition and were limited to specific classes of functions
• The Riemann integral played a crucial role in the development of modern analysis and the formalization of calculus
• Riemann's work on integration laid the foundation for further advancements in the field
  • Lebesgue's theory of measure and integration
  • Development of functional analysis and other branches of mathematics

Properties of Riemann Integrability

• A function $f(x)$ is Riemann integrable on $[a, b]$ if the limit of Riemann sums exists and is unique as the norm of the partition approaches zero
• Riemann integrability requires the function to be bounded on the interval $[a, b]$
• Continuous functions are always Riemann integrable
• Monotonic functions are Riemann integrable
• Functions with a finite number of discontinuities are Riemann integrable
  • The discontinuities must be removable or jump discontinuities
• The sum, difference, product, and quotient (when defined) of Riemann integrable functions are also Riemann integrable

Riemann Sums and Approximations

• Riemann sums approximate the integral of a function by partitioning the interval and summing the areas of rectangles
• The interval $[a, b]$ is divided into $n$ subintervals of equal width $\Delta x = \frac{b-a}{n}$
• For each subinterval $[x_{i-1}, x_i]$, a point $x_i^*$ is chosen (left endpoint, right endpoint, or midpoint)
• The Riemann sum is given by $\sum_{i=1}^n f(x_i^*) \Delta x$
• As the number of subintervals $n$ increases and the width $\Delta x$ approaches zero, the Riemann sum converges to the actual value of the integral
• Different types of Riemann sums include:
  • Left Riemann sum: $x_i^*$ is the left endpoint of each subinterval
  • Right Riemann sum: $x_i^*$ is the right endpoint of each subinterval
  • Midpoint Riemann sum: $x_i^*$ is the midpoint of each subinterval

Techniques for Evaluating Riemann Integrals

• Fundamental Theorem of Calculus: If $F(x)$ is an antiderivative of $f(x)$ on $[a, b]$, then $\int_a^b f(x) dx = F(b) - F(a)$
• Substitution method: Change of variables to simplify the integrand or transform the limits of integration
• Integration by parts: Useful for integrals involving products of functions
  • $\int_a^b u(x)v'(x) dx = [u(x)v(x)]_a^b - \int_a^b u'(x)v(x) dx$
• Partial fractions: Decomposing rational functions into simpler fractions for integration
• Trigonometric substitution: Substituting trigonometric functions to simplify the integrand
• Numerical integration methods: Approximating the integral using techniques like the Trapezoidal rule or Simpson's rule

Fundamental Theorem of Calculus

• The Fundamental Theorem of Calculus establishes a connection between differentiation and integration
• First Fundamental Theorem of Calculus: If $f(x)$ is continuous on $[a, b]$, and $F(x)$ is an antiderivative of $f(x)$, then $\int_a^b f(x) dx = F(b) - F(a)$
  • Provides a way to evaluate definite integrals using antiderivatives
• Second Fundamental Theorem of Calculus: If $f(x)$ is continuous on $[a, b]$, then $\frac{d}{dx} \int_a^x f(t) dt = f(x)$
  • Shows that differentiation and integration are inverse operations
• The Fundamental Theorem of Calculus simplifies the calculation of definite integrals and enables the development of integral calculus

Applications and Examples

• Area between curves: Calculating the area enclosed by two or more curves
  • Example: Area between $y = x^2$ and $y = x$ on the interval $[0, 1]$
• Volume of solids of revolution: Determining the volume of a solid formed by rotating a region around an axis
  • Example: Volume of a cone formed by rotating the line $y = x$ around the x-axis
• Work done by a variable force: Calculating the work done by a force that varies with position
  • Example: Work done by a spring force $F(x) = kx$ as it is compressed from $x = 0$ to $x = a$
• Average value of a function: Finding the average value of a function over an interval
  • Example: Average value of $f(x) = \sin(x)$ on the interval $[0, \pi]$
• Probability and expected value: Using integrals to calculate probabilities and expected values in continuous probability distributions
  • Example: Expected value of a continuous random variable with probability density function $f(x)$

Limitations and Extensions

• The Riemann integral has some limitations in dealing with highly oscillatory or discontinuous functions
  • Functions with infinitely many discontinuities may not be Riemann integrable (Dirichlet function)
• The Lebesgue integral extends the concept of integration to a wider class of functions
  • Lebesgue integrals can handle functions with more general types of discontinuities
  • Lebesgue integration is based on the concept of measure theory
• Improper integrals extend the Riemann integral to unbounded intervals or functions with unbounded discontinuities
  • Improper integrals are defined as limits of proper Riemann integrals
• Other extensions and generalizations of the Riemann integral include:
  • Riemann-Stieltjes integral: Integrates with respect to a monotonic function instead of the usual dx
  • Henstock-Kurzweil integral: Allows for more general partitions and tagged points in the Riemann sum
  • Gauge integral: A generalization of the Riemann and Lebesgue integrals using gauge functions