x = 1/3(y^2 + 2)^(3/2) , 0

Arc length (L) of the function x=h(y) on the interval [c,d] is given by the formula,
 L=int_c^dsqrt(1+(dx/dy)^2)dy , if x=h(y) and c <=  y <=  d,
x=1/3(y^2+2)^(3/2)
dx/dy=1/3(3/2)(y^2+2)^(3/2-1)(2y)
dx/dy=y(y^2+2)^(1/2)
Plug in the above derivative in the arc length formula,
L=int_0^4sqrt(1+(y(y^2+2)^(1/2))^2)dy
L=int_0^4sqrt(1+y^2(y^2+2))dy
L=int_0^4sqrt(1+y^4+2y^2)dy
L=int_0^4sqrt((y^2+1)^2)dy
L=int_0^4(y^2+1)dy
L=[y^3/3+y]_0^4
L=[4^3/3+4]-[0^3/3+0]
L=[64/3+4]
L=[(64+12)/3]
L=76/3
Arc length of the function over the given interval is 76/3