Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. y=3x^2, x=1, y=0, about the x-axis

Answer 1

using discs of thickness dx,
v = ∫[0,1] πr^2 dx
where r=y=3x^2
v = ∫[0,1] π(3x^2)^2 dx = 9π/5

using shells of thickness dy,
v = ∫[0,3] 2πrh dy
where r=y and h=1-x = 1-√(y/3)
v = ∫[0,3] 2πy(1-√(y/3)) dy = 9π/5

Answer 2

Vol = π ∫ y^2 dx from 0 to 1
= π ∫ 9x^4 dx from 0 to 1
= π[ (9/5)x^5 ] from 0 to 1
= π [(9/5)(1) - 0]
= 9/5π