I guess the basics steps for solving this would be:

find the area of the half-ellipse [formed by y=x^2 around y=3] = 4/3 abc

then subtract the area of the internal cone formed by the revolution of y=2x from the point of origin following the radius of x=3/2.

so the area of the cone is 1/3 (1.5)^2(3) = 2.25

now just find the area of the half-ellipse formed by y=x^2 revolved around the y axis with a bound at {y=0 and y=3}.

then subtract:

Area of half an Ellipse - 2.25

Area of the Ellipse = (4/3) (1.73)(3)(1.73) = ((4/3)(9) )/2 <--- remember its only half a ellipse

so the answer is 6 - 2.25 = 3.75 ... i think.. O and don't forget to include unit^3 in there.