Maths: Why Squaring sin x Changes Everything

Volumes of revolution take a two-dimensional area and spin it around an axis to create a three-dimensional solid. In this case, the region under y = sin x from x = 0 to x = π/3 is rotated 360° about the x-axis, producing a shape whose volume can be found by stacking infinitesimally thin circular discs. The key formula is V = π ∫ y² dx, which transforms the familiar area integral into a volume integral by squaring the function. This topic connects integration, trigonometry, and geometry in one elegant process. The area of the region uses the standard antiderivative of sin x, but the volume requires integrating sin² x. That is where the double-angle identity becomes essential: rewriting sin² x as (1 − cos 2x)/2 turns the integral into manageable terms. The final volume emerges as a combination of π² and π√3 terms, reflecting how both the constant and the trigonometric parts contribute. Understanding this relationship — between the original curve, its squared form, and the resulting solid — is central to mastering calculus applications in the IB Maths AA SL syllabus.