We compute some integrals over regions which are described using polar coordinates. Note that some of these trigonometric integrals can be tricky to work out: often a clever substitution is required. A feel for these comes with familiarity and there are useful trig identities in the log tables if you need to be reminded of them. We compute what I consider to be a difficult problem, the area of a petal of the 'rose' described by r=cos2\theta - it would be worth spending some time doodling this shape and attempting the corresponding exercises on WeBWorK or at the end of section 15.3 in Stewart.