The given parametric equations are given as:x = 3t², y = 2t³The surface generated by revolving the curve x = 3t², y = 2t³ about the y-axis can be obtained by using the formula for the surface area of revolution.Surf. area = ∫2πy √(1+(dy/dx)²)dx, where y is a function of x and x lies between the limits of rotation.The limits of rotation of the curve are 0 and 1.Therefore, the surface area of revolution is given as:Surf. area = ∫₂π 2t³ √(1+(dy/dx)²)dxWhere, x = 3t² and y = 2t³∴ dy/dx = (dy/dt)/(dx/dt) = 6t/2t² = 3/tSurface area of revolution, Surf. area = ∫₂π 2t³ √(1+(dy/dx)²)dx= ∫₂π 2t³ √(1+(3/t)²)dx= ∫₂π 2t³ √(t²+9)/t²dx= ∫₂π 2t² √(t²+9)dt= [2/3 (t²+9)³/2]₂π = (2/3)[(1²+9)³/2 - (0²+9)³/2]×(2π)Therefore, Surf. area = (2/3)[(10³/2) - (3³/2)]×(2π)= (2/3)[(100√10 - 27√2)]×(2π)≈ 560.50 sq units.Orientation of the curve is shown in the figure below:The curve is oriented in such a way that the y-axis is towards the observer and the x-axis is perpendicular to the plane of the paper.