Set up a integral and evaluate it.
y=x^8/9, y=1 and x=0 about the y-axis
assume you mean x^(8/9)
pi y^2 dy from y = 0 to y = 1
pi x^(16/9) dy
but
dy = (8/9) x^-(1/9) dx
so
(8/9)pi x^(15/9)dx from 0 to 1
(8pi/9) (9/24)x^(24/9)
(8 pi/24) = pi/3