textbook-theory · English
Difference Equations to Differential Equations — Section 4.1: The Definite Integral
Section 4.1 of Dan Sloughter's open calculus textbook introduces the definite integral via the quadrature problem — finding the area of a plane region bounded above by a continuous function and below by an interval on the x-axis. The section develops the formal definition using upper and lower Darboux sums, establishes integrability of continuous and piecewise-continuous functions, introduces left-hand and right-hand Riemann sum approximations with worked numerical examples, and covers linearity, additivity, and geometric sign conventions for the definite integral. Practice problems span approximation, geometric interpretation, and proof exercises.
definite integralRiemann sumsquadraturecalculus