29 March, 09:27
Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line.
y = 6 sin x, y = 6 cos x, 0 ≤ x ≤ π/4; about y = - 1
29 March, 10:10
Hmm well, here an example y=3,
y=3, rather than the x - x - axis.) Your integrand looks fine and reduces to
(9-18sinx+9 sin2 x) - (9-18cosx+9 cos2 x) (9-18sinx+9 sin2 x) - (9-18cosx+9 cos2 x)
= 18 (cosx-sinx) + 9 (sin2 x - cos2 x) = 18 (cosx-sinx) - 9 cos2x. = 18 (cosx-sinx) + 9 (sin2 x - cos2 x) = 18 (cosx-sinx) - 9 cos2x.
The evaluation of the volume is then
π [ 18 (sinx+cosx) - 92 sin2x ] π/4 0 π [ 18 (sinx+cosx) - 92 sin2x ] 0 π/4
= π ([ 18 (2-√ 2 + 2-√ 2) - 92 ⋅1 ] - [ 18 (0+1) - 92 ⋅0 ]) = π ([ 18 (22 + 22) - 92 ⋅1 ] - [ 18 (0+1) - 92 ⋅0 ])
= π (18 2-√ - 92 - 18) = π (18 2-√ - 452) or 9π 2 (4 2-√ - 5),
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» Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line. y = 6 sin x, y = 6 cos x, 0 ≤ x ≤ π/4; about y = - 1