Riemann Sums and Definite Integrals

By the end of this lesson, you will be able to compute left, right, and midpoint Riemann sums, set up the formal definition of the definite integral as a limit of Riemann sums, and interpret the definite integral geometrically.

The Area Problem

How do you find the area under a curve $y = f(x)$ from $x = a$ to $x = b$?

Approximate with $n$ rectangles, each of width $\Delta x = (b-a)/n$, then take the limit as $n \to \infty$.

Riemann Sums

Divide $[a,b]$ into $n$ subintervals: $x_0=a, x_1, x_2, \ldots, x_n = b$, each of width $\Delta x$.

Choose a sample point $x_i^*$ in each subinterval.

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

Left sum ($L_n$): $x_i^* = x_{i-1}$ (left endpoint)

Right sum ($R_n$): $x_i^* = x_i$ (right endpoint)

Midpoint sum ($M_n$): $x_i^* = (x_{i-1}+x_i)/2$

The Definite Integral

$$\int_a^b f(x),dx = \lim_{n \to \infty}\sum_{i=1}^{n}f(x_i^*)\Delta x$$

This limit (independent of the choice of sample points) is the definite integral of $f$ from $a$ to $b$.

Existence: if $f$ is continuous on $[a,b]$, the definite integral exists.

Example 1 — Left and Right Sums

Approximate $\int_0^2 x^2,dx$ with $n = 4$.

$\Delta x = 0.5$; left endpoints: $0, 0.5, 1, 1.5$

$L_4 = 0 + 0.25 + 1 + 2.25 = 3.5(0.5) = 1.75$

$R_4 = 0.25 + 1 + 2.25 + 4 = 7.5(0.5) = 3.75$

True value: $\int_0^2 x^2,dx = [x^3/3]_0^2 = 8/3 \approx 2.667$

As $n \to \infty$, both $L_n$ and $R_n \to 8/3$.

Geometric Interpretation

$\int_a^b f(x),dx$ = signed area between $f$ and the $x$-axis.

• Area above $x$-axis counts positively
• Area below $x$-axis counts negatively

Net signed area ≠ total area. For total area, integrate $|f(x)|$.

Example 2 — Geometric Interpretation

$$\int_0^{2\pi}\sin x,dx = 0$$

The area above the $x$-axis on $[0,\pi]$ equals the area below on $[\pi, 2\pi]$ — they cancel. The net signed area is 0.