Section 5.4: The Definite Integral

Left and Right Sums

Given a function f(x), to approximate the area bounded by the graph and the x-axis in the interval [a, b].

• Divide the interval into n subintervals of width Δx = (b - a)/n.
• The edges of the subintervals are a = x₀, x₁, x₂, ···, x_{n - 1}, x_n = b.
• The left sum is the sum of the areas of the rectangles of width Δx whose height is the value of the function at the left end of each subinterval:

  L_n = [f(x₀) + f(x₁) + ··· + f(x_{n - 1})]Δx

• The right sum is the sum of the areas of the rectangles of width Δx whose height is the value of the function at the right end of each subinterval:

  R_n = [f(x₁) + f(x₂) + ··· + f(x_n)]Δx

Error Bounds for Approximations of Area by Left or Right Sums

If f(x) > 0 and is either increasing on [a, b] or decreasing on [a, b], then:

• The total difference in height between the rectangles in the left and right sums is |f(b) - f(a)|.
• The width of the rectangles is Δx = (b - a)/n.
• The difference between the left and right sums is therefore: |f(b) - f(a)|Δx.

Since the true area lies somewhere between the left and right sums, then:

|f(b) - f(a)| · (b - a)/n

is an error bound for the approximation of the area between the graph of f and the x axis, from x = a to x = b, by L_n or R_n.

Limits of Left and Right Sums

If f(x) > 0 and is either increasing on [a, b] or decreasing on [a, b], then its left and right sums approach the same real number as n → ∞.

Limit of Riemann Sums

A Riemann sum is a more general sum that allows the height of each rectangle to be the value of the function at any point in the subinterval. If f is a continuous function on [a, b], then the Riemann sums for f on [a, b] approach a real number limit I as n → ∞.

Definite Integral

Let f be a continuous function on [a, b]. The limit I of Riemann sums for f on [a, b], guaranteed to exist by Theorem 3, is called the definite integral of f from a to b and is denoted as:

∫ab f(x) dx

The integrand is f(x), the lower limit of integration is a, and the upper limit of integration is b.

Properties of Definite Integrals

1. ∫aa f(x) dx = 0
2. ∫ab f(x) dx = - ∫ba f(x) dx
3. ∫ab kf(x) dx = k ∫ab f(x) dx
4. ∫ab [f(x) ± g(x)] dx = ∫ab f(x) dx ± ∫ab g(x) dx
5. ∫ac f(x) dx = ∫ab f(x) dx + ∫bc f(x) dx