Let R be the region bounded by the curve y=cos(x)sin(x) and the x-axis between x

Question

Let R be the region bounded by the curve y=cos(x)sin(x) and the x-axis between x=0 and x= Pi/2. Let S be the solid obtained by rotating R about the x-axis. Find the volume of S

Explanation / Answer

Region R : {(x,y) | 0 = x = vp ; 0 = y = sin x² } ; V = 2p ?(x = 0 to vp) x sin²x dx = ...( cos 2x = 1 - 2sin²x ==> sin²x = (1 - cos 2x)/2 ) ... = 2p ?(x = 0 to vp) x (1 - cos 2x)/2 dx = = p ?(x = 0 to vp) x - x cos 2x) dx = = px²/2 | (x = 0 to vp) - p ?(x = 0 to vp) x cos 2x dx = ... u = x ; du = dx ... dv = cos 2x dx ; v = 1/2 sin 2x .. = p²/2 - p/2 x sin 2x | (x = 0 to vp) + p/2 ?(x = 0 to vp) sin 2x dx = = p²/2 - pvp /2 sin 2vp - p/4 cos 2x | (x = 0 to vp) = = p²/2 - pvp /2 sin 2vp - p/4 cos 2vp + p/4 = ... - p/4 cos 2vp = - p/4 ( 1 - 2 sin² vp ) = - p/4 + p/2 sin² vp ... = p²/2 - pvp /2 sin 2vp + p/2 sin² vp ˜ 7.53

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