Find the volume formed by rotating the region about the y-axis?

Question

Find the volume formed by rotating the region enclosed by:

x=10y and y^3=x with y>or+0

about the y-axis

Answer 1

Greetings,

The points of intersection are

y^3 = 10y

y^3 - 10y = 0

y(y^2 - 10) = 0

y = 0, x = 0 or y = sqrt10, x = 10sqrt10

Volume of revolution between these curves is

V = pi∫(10y)^2 - (y^3)^2 dy from 0 to sqrt10

= pi [100y^3/3 - y^7/7] from 0 to sqrt10

= pi*y^3[100 - y^4/7]

= pi*(sqrt10)^3*600/7

= 8515.36

Regards