calculus problem?

所求旋轉體積
= ∫ π*∣y∣^2 dx , from x = 0 to x = π/2
= ∫ π*∣ sin( x + π/4 ) ∣^2 dx
= π * ∫ sin^2 ( x + π/4 ) dx
= π * ∫ (1/2)*[ 1 - cos( 2x + π/2 ) ] dx , 利用半角公式 sin (β/2) = ± √[ ( 1 - cos β ) / 2 ]
= π * ∫ [ 1/2 - (1/2)*cos( 2x + π/2 ) ] dx
= π * [ (1/2)x - (1/4)*sin( 2x + π/2 ) ] , from x = 0 to x = π/2
= π * { [ (1/2)(π/2) - (1/4)*sin 3π/2 ] - [ - (1/4)*sin π/2 ] }
= π * [ π/4 - (1/4)*(-1) + 1/4 ]
= (1/4)π^2 + (1/2)π ..... Ans