Example 2 the arc of the parabola y = 3x2 from (5, 75) to (10, 300) is rotated about the y-axis. find the area of the resulting surface. solution 1 using y = 3x2 and dy dx = we have, from this formula, s = 2πx ds = 10 2πx 1 + dy dx 2 dx 5 = 2π 10 x 1 + 36x2 dx 5 . substituting u = 1 + 36x2, we have du = dx. remembering to change the limits of integration, we have s = π 36 3601 u du 901 = π 36 3601 901 = . solution 2 using x = y 3 and dx dy = we have s = 2πx ds = 300 2πx 1 + dx dy 2 dy 75 = 2π 300 y 3 1 + 1 12y dy 75 = π 3 300 12y + 1 dy 75 = π 36 3601 u du 901 (where u = 1 + 12y) = (as in solution 1)