It is examinable for the HSC and it's in a few of the past HSC papers (98, I think? can't remember).
Integration from first principles is not often taught, and it's only been asked twice - once in an HSC paper and once in a Catholic trial paper. Both questions involved the integral in the example I posted above, where you were given the equation of a semicircle. You can't integrate it normally, because the function under the square root sign is not linear, but you can use the formula for the area of a circle to calculate an equivalent answer.