Syllabus
Calculus II
This course develops the integral calculus, from its definition as a limit of finite sums through to its use in solving geometric and physical problems. It follows on from differential calculus and assumes familiarity with limits, derivatives, and the elementary functions.
The course has three movements. We begin by building the integral from first principles and establishing the Fundamental Theorem of Calculus, the result that links differentiation and integration and makes the whole subject tractable. We then assemble a toolkit of integration techniques, extending our range of functions to include logarithms, inverse trigonometric and hyperbolic functions along the way. Finally, we put those tools to work: computing areas, volumes, arc lengths and surface areas, and reformulating problems in polar coordinates where the geometry makes it natural.
A note on sequence: logarithmic and exponential functions are defined properly in Chapter 7, where the natural logarithm is introduced as an integral. Examples involving them are therefore set aside until that point.
What you will be able to do
By the end of the course you should be able to evaluate definite and indefinite integrals using a range of techniques, recognise which technique a given integral calls for, handle limits of indeterminate form and integrals with infinite range or unbounded integrands, and apply integration to compute areas, volumes of revolution, arc lengths and surface areas in both Cartesian and polar coordinates.