Let f be a continuous function on [0, 1], which is bounded below by 1, but is not identically 1. Let R be the region in the plane given by 0 ≤ x ≤ 1, 1 ≤ y ≤ f(x). Let
R_1 = \{(x, y) ∈ R|y ≤ \bar{y}\}~~~and~~~R_2 = \{(x, y) ∈ R|y ≥\bar{ y}\},
where \bar{y} is the y-coordinate of the centroid of R. Can the volume obtained by rotating R_1 about the x-axis equal that obtained by rotating R_2 about the x-axis?