The given curve is rotated about the y-axis. Find the area of the resulting surface. y = 5 - x^2, 0 = x = 3 Solution we get a paraboloid when we rotate y=5-x^2 when we consider an elemental ring on the surface it has radius r=x,and length dy so the area of elemental ring ds=2*pi*r*dl=2*pi*x*|dy| we have dy= -2xdx so ds=4*pi*x^2 dx integrating form 0 to 3 gives surface area =4/3 * pi *(27-0)=36*pi.

1. The given curve is rotated about the y-axis. Find the area of the resulting surface.
y = 5 - x^2, 0 = x = 3
Solution
we get a paraboloid when we rotate y=5-x^2 when we consider an elemental ring on
the surface it has radius r=x,and length dy so the area of elemental ring
ds=2*pi*r*dl=2*pi*x*|dy| we have dy= -2xdx so ds=4*pi*x^2 dx integrating form 0 to 3 gives
surface area =4/3 * pi *(27-0)=36*pi