Find the area of the surface obtained by rotating the curve

Find the area of the surface obtained by rotating the curve x= y1/2 - 1/3y3/2 from y=1 to y=3 about the y-axis. .852

Solution

dx/dy = 1/2 y^-1/2 - 1/2 y^1/2
Surface area,S = 2 integral of x 1+(dx/dy)^2 dy....limit from 1 t0 3
S = 2 integral of (y^1/2 - 1/3 y^3/2)1+[1/2 y^-1/2 - 1/2 y^1/2]^2
S = 2 integral of (y^1/2 - 1/3 y^3/2)[1+ 1/4 y^-1 - 1 + 1/4 y]
S = integral of (y^1/2 - 1/3 y^3/2)(1/y + y)
S = integral of (y^1/2 - 1/3 y^3/2)/y^1/2(1 + y^2)
S = integral of (1 - y/3)(1 + y^2)dy
S = integral of ((1 + y^2) - 1/3 y(1 + y^2)dy
integrating wrt y
S = [y/2 1+y^2 + 1/2 ln|y+1+y^2| - 2/9 (1+y^2)^3/2]
apply limits
S = [3/2 10 + 1/2 ln|3 + 10| - 2/9 10.10] - [1/2 2 +1/2ln|1+2| - 2/9 22]