2005 Extension 2 HSC Volumes Question

The 2005 Extension 2 HSC contains a nice volume by cylindrical shells question.
What makes it even more interesting is the resulting answer.  The volume is finite.  All well and good, but this is a variation of the famous question often seen in a multi-variable calculus course, known as Gabriel's Trumpet  or the Infinite Paint Can.  Such a shape would have a finite volume, but an infinite surface area.  It wouldn't be able to hold enought paint to paint the outside of the can!

2011 HSC Integration Questions

The 2011 HSC has some nice integration questions;  They start out fairly basic in question 1, progressing to a good one in Question 7.
Both of these questions are included here for completeness, but don't worry too much about Question 7 - it contains a property of integrals we haven't studied yet.  That being said, it won't hurt to have a look at it!

100 Integrals from Coroneos

One of the good 4 unit books around is the one by Jim Coroneos.  It was published years ago, and still contains some really good questions.  In particular, at the end of the integration chapter, there are 100 integrals.  If you can do these, then you are well on your way to a good understanding.
I've simply re-typeset them, and you can get them below.

Partial Fractions

Partial fractions plays a major role in the theory of integration, and is examined in all HSC papers.  You should be able to identify the way to decompose your function, but there has been a trend over the years to give you the decomposition and ask you to find the missing numbers.  The second part usually involves the integration.

Here are two questions on partial fractions from past HSC papers.

Reduction Formulae

As we move into the integration topic, one application of integration by parts is to reduction (or recurrence) formulae.  The two presentations are from past HSC papers.

It should also be noted that while most reduction formulae are completed using integration by parts, there are some that are achieved through use of the Pythagorean Identities...I'll post some more on this later!

2001 Trial HSC - International Grammar

Below is question 4 from the 2001 International Grammar Trial HSC. It has some good questions on polynomials.

A nice question from the latter stages of the Extension 2 paper.It shows how seamlessly complex numbers and polynomials go together!