Riemann Sums and Area Approximation: Foundations of Integration

Riemann Sums and Area Approximation

Introduction to Area Approximation

In calculus, the area under a curve can be approximated using sums of rectangles. This process forms the basis for definite integrals and is known as the Riemann Sum approach. The region R is typically bounded by a function, the x-axis, and vertical lines at specified points.

• Region R: The area bounded by $y = x^2$, $y = 0$, $x = 1$, and $x = 3$.

• Goal: Approximate the area of R using right endpoint Riemann sums.

Partitioning the Interval

To approximate the area, the interval $[1, 3]$ is divided into n subintervals of equal width.

• Subintervals: Each has width $\Delta x = \frac{2}{n}$, since $3 - 1 = 2$.

• Endpoints: The $i$-th subinterval is $[x_{i-1}, x_i]$ where $x_i = 1 + \frac{2i}{n}$ for $i = 0, 1, ..., n$.

Right Endpoint Approximation

For the right endpoint approximation, the height of each rectangle is determined by the function value at the right endpoint of each subinterval.

• Area Approximation Formula:

$R_n = \sum_{i=1}^{n} f(x_i) \Delta x$

• For $f(x) = x^2$:

$R_n = \sum_{i=1}^{n} \left(1 + \frac{2i}{n}\right)^2 \cdot \frac{2}{n}$

Expanding the Sum

Expand the squared term and simplify:

$\left(1 + \frac{2i}{n}\right)^2 = 1 + \frac{4i}{n} + \frac{4i^2}{n^2}$

So,

$R_n = \sum_{i=1}^{n} \left[1 + \frac{4i}{n} + \frac{4i^2}{n^2}\right] \cdot \frac{2}{n}$

Distribute $\frac{2}{n}$:

$R_n = \sum_{i=1}^{n} \frac{2}{n} + \sum_{i=1}^{n} \frac{8i}{n^2} + \sum_{i=1}^{n} \frac{8i^2}{n^3}$

Evaluating the Sums

Use formulas for sums of integers and squares:

• $\sum_{i=1}^{n} 1 = n$

• $\sum_{i=1}^{n} i = \frac{n(n+1)}{2}$

• $\sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6}$

Substitute into $R_n$:

$R_n = \frac{2}{n} \cdot n + \frac{8}{n^2} \cdot \frac{n(n+1)}{2} + \frac{8}{n^3} \cdot \frac{n(n+1)(2n+1)}{6}$

Simplify:

$R_n = 2 + 4 \cdot \frac{n+1}{n} + \frac{4}{3} \cdot \frac{(n+1)(2n+1)}{n^2}$

Taking the Limit as $n \rightarrow \infty$

To find the exact area, take the limit as $n$ approaches infinity:

$\lim_{n \rightarrow \infty} R_n = 2 + 4 \cdot 1 + \frac{4}{3} \cdot 2 = 2 + 4 + \frac{8}{3} = \frac{20}{3}$

Therefore, the exact area under $y = x^2$ from $x = 1$ to $x = 3$ is $\frac{20}{3}$.

General Definition of Riemann Sums

A Riemann Sum is a sum of the form:

$\sum_{i=1}^{n} f(x_i^*) \Delta x_i$

• $x_i^*$ is a sample point in the $i$-th subinterval $[x_{i-1}, x_i]$.

• $\Delta x_i = x_i - x_{i-1}$ is the width of the $i$-th subinterval.

• Subintervals may have equal or unequal widths.

Partitions and Mesh

A partition $P$ of $[a, b]$ divides the interval into subintervals specified by points $a = x_0 < x_1 < \ldots < x_n = b$.

• Even Partition: All subintervals have equal width $\Delta x = \frac{b-a}{n}$.

• Mesh: The mesh of a partition is the length of the largest subinterval: $\max \{ \Delta x_i \}$.

Types of Riemann Sums

• Left Endpoint: Use $x_{i-1}$ for the height.

• Right Endpoint: Use $x_i$ for the height.

• Midpoint: Use $\frac{x_{i-1} + x_i}{2}$ for the height.

Example Table: Types of Riemann Sums

Type	Sample Point	Formula
Left Endpoint	$x_{i-1}$	$\sum_{i=1}^{n} f(x_{i-1}) \Delta x$
Right Endpoint	$x_i$	$\sum_{i=1}^{n} f(x_i) \Delta x$
Midpoint	$\frac{x_{i-1} + x_i}{2}$	$\sum_{i=1}^{n} f\left(\frac{x_{i-1} + x_i}{2}\right) \Delta x$

Properties and Applications

• Riemann Sums provide a foundation for the definition of the definite integral.

• As the number of subintervals increases (and their width decreases), the approximation improves.

• The definite integral is defined as the limit of the Riemann sum as the mesh approaches zero.

Definite Integral:

$\int_a^b f(x) dx = \lim_{\text{mesh} \rightarrow 0} \sum_{i=1}^{n} f(x_i^*) \Delta x_i$

Summary

• Riemann sums approximate areas under curves by summing rectangles.

• Different types of sums (left, right, midpoint) use different sample points.

• As the partition becomes finer, the sum approaches the exact area, which is the definite integral.

Additional info: Some context and formulas were inferred for completeness, including the explicit calculation of the area and the general definition of Riemann sums.