Related rates

Integration by substitution (5.3)

Differential equations (5.4)

Area under the graph of a positive function. Riemann sums.

Definition of a Riemann integrable function on [a,b].

Theorem: a continuous function on a closed interval is Riemann integrable.

Simple examples of functions not continuous, yet integrable.

The definite integral of f(x) on [a,b] (the Riemann integral) is the area between the graph of f and the segment [a,b] with sign.

Calculation of areas as limits of Riemann sums.

And the other way around: calculation of limits of sums noting that they are Riemann sums.

The Fundamental Theorem of Calculus.

Properties of definite integrals.

Area between two curves.

Volumes of solids of revolution: disk (or washer) method, cylindrical shells method.

Volumes by integrating the areas of the cross-sections.

Arc length.

Area of a surface of revolution.

e^x and ln: they are inverse to each other, limits for e, their domains, range, derivatives, graphs,

integrating 1/x, and e^x (as such or in combination with substitutions)