In this, the fourth section of the course, the topics include:
Interpretations and properties of definite integrals • Definite integral as a limit of Riemann sums. • Definite integral of the rate of change of a quantity over an interval interpreted as the change of the quantity over the interval. • Basic properties of definite integrals (examples include additivity and linearity).
Fundamental Theorem of Calculus • Use of the Fundamental Theorem to evaluate definite integrals. • Use of the Fundamental Theorem to represent a particular antiderivative and the analytical and graphical analysis of functions so defined.
Techniques of antidifferentiation • Antiderivatives following directly from derivatives of basic functions. • Antiderivatives by substitution of variables (including change of limits for definite integrals).
Numerical approximations to definite integrals. • Use of Riemann sums (using left, right, and midpoint evaluation points) and trapezoidal sums to approximate definite integrals of functions represented algebraically, graphically, and by tables of values.