Find the volume of the solid obtained by rotating the region bounded by the curve y = sin ( 6 x 2 ) and the x -axis, 0 ≤ x ≤ √ π 6 , about the y -axis.

Answer 1

v = 2π∫₀^√π/6 xsin(6x²)dx = - π/6cos(6x²)₀ ^√π/6 = - π/6cosπ + π/6cos0 = π/3

Answer 2

The volume can be found using the shell method. A vertical strip of the region at a distance x from the y-axis has height sin(6x²), so the volume is ∫ (2π)(radius)(height)(thickness) = ∫[0, √π/6] 2π x sin(6x²) dx. Letting u = 6x², we get du = 12x dx or (1/6)du = 2x dx; thus

∫[0, √π/6] 2π x sin(6x²) dx = ∫[0, π] π/6 sin u du

.........................................= π/6 [-cos π - (-cos 0)] = π/6 (1 - (-1))

.........................................= 2π/6 = π/3