Final answer:
To find the equation for the surface obtained by rotating the parabola y = x² about the y-axis, we can use the method of cylindrical shells.
Step-by-step explanation:
To find the equation for the surface obtained by rotating the parabola y = x² about the y-axis, we can use the method of cylindrical shells. In this method, we consider infinitesimally thin cylinders parallel to the y-axis, each with a radius equal to the x-coordinate of a point on the parabola and a height equal to the y-coordinate of that same point.
The volume of each cylinder is given by the formula V = 2πx * y * dx, where dx is the infinitesimal thickness of the cylinder. We integrate this volume from x = 0 to x = c, where c is the x-coordinate of the point where the parabola intersects the y-axis.
After integrating, we get the equation for the surface as y = (1/2)πc²x - (1/3)πc³.