Need help with volumes of solids of revolution?

Question

Find the volume of the solid generated when the region in the first quadrant, bounded by the x axis, the line x=8 and the curve y=x^1/3 is rotated about:

i) the x axis

ii) the y axis

Answer 1

i)Radius of the curve is x^1/3

Volume of the solid is V=∫πr^2dx

                                              8

                                         =  ∫π(x^1/3)^2dx

                                            0

                                            8

                                         =  ∫π(x^2/3)dx

                                            0 

                                                                   8

                                         =π(x^5/3/(5/3))

                                                                  0

                                         =3π/5(8^5/3-0)

                                         =3π/5(32)

                                         =96π/5  cubic units

Answer 2

ii)Radius of the curve is

                       y=x^1/3

                       x=y^3

Radius of the curve is y^3

Limits :x=y^3

              8=y^3

              y=2

Volume of the solid is V=∫πr^2dy

                                              2

                                         =  ∫π(y^3)^2dy

                                            0

                                            2

                                         =  ∫π(y^6)dy

                                            0 

                                                                 2

                                         =π(y^7/(7))

                                                                  0

                                         =π/7(2^7-0)

                                         =128π/7  cubic units