Riemann sums are a powerful tool for approximating definite integrals. By dividing an interval into subintervals and summing the areas of rectangles, we can estimate the area under a curve. This method forms the foundation for understanding the Riemann integral.

As we increase the number of subintervals, our approximation becomes more accurate. This concept leads to the definition of the definite integral as the limit of Riemann sums. Understanding Riemann sums is crucial for grasping integration and its applications in calculus.

Approximating Definite Integrals

Riemann Sums

• Riemann sums approximate the definite integral of a function over a closed interval $[a, b]$
• The interval $[a, b]$ is partitioned into $n$ subintervals of equal width $\Delta x = (b - a) / n$
• The left Riemann sum uses the left endpoint of each subinterval to calculate the rectangle's height
• The right Riemann sum uses the right endpoint of each subinterval to calculate the rectangle's height
• The midpoint Riemann sum uses the midpoint of each subinterval to calculate the rectangle's height, providing a more accurate approximation than left or right Riemann sums (trapezoidal rule)

Calculating Riemann Sum Approximations

• The Riemann sum approximation is calculated by summing the areas of the rectangles: $\sum f(x_i) \Delta x$, where $x_i$ is the chosen point (left, right, or midpoint) in the $i$-th subinterval
• As the number of subintervals $n$ increases and the width $\Delta x$ approaches zero, the Riemann sum approximation becomes more accurate
  • Example: Approximating the area under the curve $f(x) = x^2$ on the interval $[0, 1]$ using a left Riemann sum with $n = 4$ subintervals
  • Example: Comparing the accuracy of left, right, and midpoint Riemann sums for the function $f(x) = \sin(x)$ on the interval $[0, \pi]$ with increasing values of $n$

Riemann Sums and Definite Integrals

Relationship between Riemann Sums and Definite Integrals

• The definite integral of a function $f(x)$ over the interval $[a, b]$ is the limit of Riemann sums as the number of subintervals $n$ approaches infinity and the width $\Delta x$ approaches zero
• The definite integral represents the exact area under the curve $f(x)$ over the interval $[a, b]$, while Riemann sums provide approximations of this area
• If $f(x)$ is continuous on $[a, b]$, the limit of the Riemann sums (left, right, or midpoint) as $n \to \infty$ equals the definite integral of $f(x)$ from $a$ to $b$
  • Example: Showing that the limit of the midpoint Riemann sums for $f(x) = x^2$ on $[0, 1]$ equals the definite integral $\int_0^1 x^2 dx$

Fundamental Theorem of Calculus

• The Fundamental Theorem of Calculus establishes the connection between the definite integral and antiderivatives
• It provides a way to evaluate definite integrals without using Riemann sums
• The theorem states that if $F(x)$ is an antiderivative of $f(x)$ on $[a, b]$, then $\int_a^b f(x) dx = F(b) - F(a)$
  • Example: Using the Fundamental Theorem of Calculus to evaluate the definite integral $\int_0^1 x^2 dx$ by finding an antiderivative of $f(x) = x^2$

Error in Riemann Sum Approximations

Bounding the Error

• The error in a Riemann sum approximation is the difference between the actual value of the definite integral and the approximation
• For a continuous function $f(x)$ on $[a, b]$, the error in a Riemann sum approximation is bounded by the maximum and minimum values of $f(x)$ on each subinterval
• The error in a left Riemann sum is bounded by the area between the curve and the upper sum (if $f$ is increasing) or the lower sum (if $f$ is decreasing)
• The error in a right Riemann sum is bounded by the area between the curve and the lower sum (if $f$ is increasing) or the upper sum (if $f$ is decreasing)
  • Example: Bounding the error in a left Riemann sum approximation for $f(x) = \sin(x)$ on $[0, \pi/2]$ with $n = 8$ subintervals

Reducing the Error

• The midpoint Riemann sum generally provides a more accurate approximation than left or right Riemann sums, as it tends to balance out the overestimations and underestimations
• The error in a Riemann sum approximation decreases as the number of subintervals $n$ increases and the width $\Delta x$ approaches zero
  • Example: Comparing the error in left, right, and midpoint Riemann sum approximations for $f(x) = e^x$ on $[0, 1]$ as $n$ increases

Partition Strategies for Improved Approximations

Non-uniform Partitions

• Non-uniform partitions can be used to improve the accuracy of Riemann sum approximations, particularly when the function has rapid changes or irregularities in certain regions
• Adaptive quadrature methods, such as Simpson's rule or Gaussian quadrature, use non-uniform partitions and higher-order approximations to achieve better accuracy with fewer subintervals
  • Example: Using a non-uniform partition to approximate the integral of $f(x) = 1/(1+x^2)$ on $[0, 1]$, with more subintervals concentrated near $x = 0$ where the function changes rapidly

Higher-order Approximations

• Simpson's rule approximates the integral using quadratic polynomials on each subinterval, providing a higher-order approximation than Riemann sums
• Gaussian quadrature uses optimal partition points (called Gaussian nodes) and weights to minimize the approximation error for a given number of subintervals
• Composite quadrature methods, such as the composite Simpson's rule or composite Gaussian quadrature, divide the interval into smaller subintervals and apply the quadrature method on each subinterval to improve the overall approximation accuracy
  • Example: Approximating the integral of $f(x) = \cos(x)$ on $[0, \pi]$ using Simpson's rule with $n = 6$ subintervals
  • Example: Comparing the accuracy of composite Simpson's rule and composite Gaussian quadrature for the integral of $f(x) = \sqrt{1-x^2}$ on $[-1, 1]$

Adaptive Quadrature Methods

• Adaptive quadrature methods can dynamically adjust the partition based on the local behavior of the function
• These methods allocate more subintervals to regions with rapid changes or irregularities to maintain a desired level of accuracy
  • Example: Implementing an adaptive quadrature method to approximate the integral of $f(x) = e^{-x^2}$ on $[-3, 3]$, with a higher concentration of subintervals near $x = 0$