Premed · Premed · Calculus 1
Lecture 19: The Definite Integral and Riemann Sums
Calculus I — Differential Calculus
Learning Objectives
By the end of this lecture, students will be able to:
- Approximate the area under a curve using Riemann sums (left, right, midpoint)
- Express the definite integral as the limit of a Riemann sum
- Interpret the definite integral as a signed area
- State and apply basic properties of the definite integral
- Evaluate simple definite integrals using geometry
Lecture Content
I. The Area Problem
The central goal of this lecture is to find the area of the region bounded by y = f(x), the x-axis, and the vertical lines x = a and x = b, where f(x) >= 0. The strategy is to approximate the area using rectangles and then take a limit as the rectangles become infinitely thin.
II. Riemann Sums
To construct a Riemann sum, partition the interval [a, b] into n subintervals of equal width Delta x = (b - a)/n. The subintervals are [x_0, x_1], [x_1, x_2], ..., [x_{n-1}, x_n], where x_i = a + i Delta x. In each subinterval [x_{i-1}, x_i], choose a sample point x_i. The Riemann sum is then S_n = sum_{i=1}^{n} f(x_i) Delta x.
The most common choices for sample points yield three standard sums. The Left Riemann sum uses x_i = x_{i-1}, the left endpoint of each subinterval. The Right Riemann sum uses x_i = x_i, the right endpoint. The Midpoint Riemann sum uses x_i* = (x_{i-1} + x_i)/2, the midpoint. For an increasing function, the left sum underestimates and the right sum overestimates the true area, while the midpoint sum typically provides a better approximation.
<image>Three panels showing Riemann sums for f(x) = x^2 on [0, 2] with n = 5 rectangles. Panel A: Left Riemann sum — rectangles using left endpoints, area underestimates the true area (for increasing functions). Panel B: Right Riemann sum — rectangles using right endpoints, area overestimates. Panel C: Midpoint Riemann sum — rectangles using midpoints, gives a better approximation. The actual curve y = x^2 is drawn in each panel with the rectangles shaded. The approximate area values are displayed for each. Title: "Left, Right, and Midpoint Riemann Sums."</image>
III. The Definite Integral as a Limit
The definite integral of f from a to b is defined as integral from a to b of f(x) dx = lim_{n -> infinity} sum_{i=1}^{n} f(x_i) Delta x, provided this limit exists and gives the same value regardless of how the sample points are chosen. If f is continuous on [a, b], or if f is bounded with only finitely many discontinuities, then the definite integral exists and f is said to be integrable.
Unlike the indefinite integral, which produces a family of functions, the definite integral is a number. The notation integral from a to b of f(x) dx uses a as the lower limit and b as the upper limit of integration.
IV. Signed Area Interpretation
The definite integral represents signed area. Where f(x) > 0, the area between the curve and the x-axis counts as positive (above the x-axis). Where f(x) < 0, the area counts as negative (below the x-axis). The definite integral therefore equals the area above the x-axis minus the area below it. To compute the total (unsigned) area between the curve and the x-axis, one must integrate the absolute value: integral from a to b of |f(x)| dx.
<image>A graph of a function f(x) that is positive on [a, c] and negative on [c, b]. The region above the x-axis on [a, c] is shaded green and labeled "+" (positive area). The region below the x-axis on [c, b] is shaded red and labeled "-" (negative area). The formula: definite integral = (green area) - (red area) is displayed. Title: "The definite integral as signed area."</image>
V. Properties of the Definite Integral
Several properties make working with definite integrals systematic. Linearity ensures that the integral of a sum is the sum of the integrals, and constants factor out: integral from a to b of [f(x) + g(x)] dx = integral from a to b of f(x) dx + integral from a to b of g(x) dx, and integral from a to b of c f(x) dx = c integral from a to b of f(x) dx.
Additivity over intervals states that integral from a to b of f(x) dx + integral from b to c of f(x) dx = integral from a to c of f(x) dx. Reversal of limits introduces a sign change: integral from b to a of f(x) dx = - integral from a to b of f(x) dx. Over a zero-width interval, integral from a to a of f(x) dx = 0.
The comparison properties are also important. If f(x) >= 0 on [a, b], then the integral is non-negative. If f(x) >= g(x) on [a, b], then the integral of f is at least as large as the integral of g. And if m <= f(x) <= M on [a, b], then m(b - a) <= integral from a to b of f(x) dx <= M(b - a).
VI. Evaluating Integrals Using Geometry
Some definite integrals can be computed by recognizing the region as a familiar geometric shape. For integral from 0 to 3 of (2x + 1) dx, the region is a trapezoid with parallel sides of height 1 (at x = 0) and 7 (at x = 3) and width 3, giving area (1/2)(1 + 7)(3) = 12.
For integral from -2 to 2 of sqrt(4 - x^2) dx, the integrand describes the upper half of a circle with radius 2, so the area is (1/2) pi 4 = 2*pi. For integral from -1 to 1 of |x| dx, the region consists of two right triangles, each with base 1 and height 1, giving a total area of 1.
VII. Sigma Notation and Computing Riemann Sums
Several summation formulas are useful when evaluating Riemann sums directly: sum_{i=1}^{n} i = n(n+1)/2, sum_{i=1}^{n} i^2 = n(n+1)(2n+1)/6, and sum_{i=1}^{n} i^3 = [n(n+1)/2]^2.
As an example, evaluate integral from 0 to 1 of x^2 dx using the limit definition. With Delta x = 1/n and right endpoints x_i = i/n, the Riemann sum is S_n = sum_{i=1}^{n} (i/n)^2 (1/n) = (1/n^3) sum_{i=1}^{n} i^2 = (1/n^3) * n(n+1)(2n+1)/6 = (n+1)(2n+1)/(6n^2). Taking the limit as n approaches infinity gives 2/6 = 1/3, confirming that integral from 0 to 1 of x^2 dx = 1/3.
<image>A visualization of the Riemann sum computation for integral from 0 to 1 of x^2 dx. Four sub-panels showing n = 4, 8, 16, and 32 right-endpoint rectangles under the parabola y = x^2 on [0, 1]. As n increases, the rectangles more closely fill the area under the curve. Below each panel, the Riemann sum value is displayed (e.g., S_4 = 0.46875, S_8 = 0.3984, S_16 = 0.3652, S_32 = 0.3496), converging toward 1/3 ≈ 0.3333. Title: "Riemann sums converging to the definite integral."</image>
