A level Maths question - The graph of y=2sin(2x)+1 is rotated 360 degrees about the x-axis to form a solid. Find the volume enclosed by the curve, the co-ordinate axes and the line x=pi/2

First, recall that the formula for finding such a volume is pi times the integral from 0 to pi/2 of y^2 with respect to x. This leads to pi times the integral from 0 to pi/2 of 4sin^2(2x) + 4sin(2x) + 1 (this is obtained by expanding the expression for y). The last two terms of this expression are easily integrable using standard results learned during A level, whilst the first term will require some more thought.Considering the trigonometric identity cos(2A) = cos^2(A)-sin^2(A) = 1 - 2sin^2(A), the first term 4sin^2(2x) is evidently equivalent to 2 - 2cos(4x). Thus the integral becomes the integral with respect to x of 3 + 4sin(2x) - 2cos(4x) from 0 to pi/2. Using the standard results for basic integrals this becomes the definite integral pi*[3x-2cos(2x)-1/2sin(4x)] with the same limits as before. This can be evaluated quite simply by substituting in the limits as required, and recalling the values of trig functions for some standard angles. The correct result is pi*(3pi+8)/2.