Question

5. Find the volume of the solid generated by revolving the region bounded by y = x2, y = 0 and x = 1 about
(a) the x-axis
(b) the y-axis

Answer 1

(c) theline x=2
(d) the line y = −2

Answer 2

I'll do (a) and you can apply the logic to the others.

Using discs, the small discs have thickness dx
v = ∫[0,1] πr^2 dx
where r=y=x^2
v = π∫[0,1] x^4 dx
= π * 1/5 x^5 [0,1]
= π/5

Using shells, the thickness is dy, so we have
v = ∫[0,1] 2πrh dy
where r=y and h=(1-x)=(1-√y)
v = 2π∫[0,1] y(1-√y) dy
= 2π (1/2 y^2 - 2/5 y^5/2) [0,1]
= 2π (1/2 - 2/5)
= π/5

For rotation about the line y=-2,
discs have a hole in them
shells have a radius y+2

Give it a shot; try both ways to validate your answers