StellarRiemann sums can be written compactly using summation notation. If an interval [a,b] is divided into n subintervals of equal width, then Δ x=(b-a)/n and a typical sample point is often written as xi. A general Riemann sum has the form . This notation is important because it expresses the total approximation as a structured sum and prepares you for the idea of taking a limit as the number of rectangles increases without bound. The definite integral notation is the limit of such sums when that limit exists.
The symbol represents continuous summation, the numbers a and b are the limits of integration, f(x) is the integrand, and dx indicates the variable of integration. You should understand that a definite integral is not just a new symbol but a formal limit of Riemann sums. This is one of the most important transitions in the course: the approximation process becomes an exact calculus object. Even before evaluating integrals exactly, you should be able to connect rectangle sums, sigma notation, and integral notation as three views of the same underlying idea of accumulated change.