6.3 Riemann Sums & Integral Notation

From sum to integral

As the number of rectangles → ∞ (widths → 0), the Riemann sum becomes the definite integral: ∫[a to b] f(x) dx = limit of Riemann sums.

Notation

In ∫[a to b] f(x) dx: a is the lower limit, b the upper limit of integration, f(x) is the integrand, and dx shows the variable. The definite integral is the (net) area between f and the x-axis from a to b.

Key terms

Definite integral

The limit of Riemann sums as widths → 0.

∫[a to b] f(x) dx

Net area between f and the x-axis from a to b.

Limits of integration

a (lower) and b (upper).

Integrand

The function f(x) being integrated.

Sum → integral

Infinitely many thin rectangles give the exact area.

Try an AP-style question

Which integral equals lim(n→∞) Σᵢ₌₁ⁿ (1 + 2i/n)² · (2/n)?

1. ∫₁² x² dx
2. ∫₁³ x² dx
3. ∫₀² x² dx
4. ∫₀³ x² dx