Find the volume of the solid obtained by rotating the region

Question

 Find the volume of the solid obtained by rotating the region bounded by the given curves below about y=-4. y=0, y=cos(3x), x=pi/6, x=0

Answer 1

Set z = y + 4 = 4 + cos x and use the shell method to revolve the area about the x axis.

Answer 2

for y= -4 the bounded region is y= cos(3 x) +4, y=4 x ranges from 0 to π/6

V= π∫₀^π/6 [( cos(3 x) +4)² - 16]dx V= π ∫₀^π/6 [ (1+cos(6 x))/2 + 8 cos(3 x) ] dx V= π [ x/2 + sin(6 x)/12 +(8/3) sin(3 x)] |₀^π/6 V= π²/12 +8π/3

for y=0 the bounded region is y= cos(3 x) y=0 V= π[ ∫₀^π/6 [cos²(3 x) -0]dx =π[ x/2 + sin(6 x)/12] |₀^π/6 =π²/12