Please go through the process in DETAIL I am having trouble

Please go through the process in DETAIL. I am having trouble going through the integral process and want to see it done step-by-step. Thank you.

Solution

First we find x-coordinates of points of intersection of y = 2 + x - x² y = 2 - x 2 - x = 2 + x - x² x² - 2x = 0 x (x - 2) = 0 x = 0, x = 2 So we integrate from x = 0 to x = 2 On this interval 2 + x - x² > 2 - x Axis of rotation = y-axis (line x = 0) Surface area of cylindrical shell = 2p r h where r = distance from axis of rotation = x and h = height of cylinder = (2 + x - x²) - (2 - x) = 2x - x² V = 2p ?0² r h dx V = 2p ?0² x (2x - x²) dx V = 2p ?0² (2x² - x³) dx Solving, we get V = 8p/3 =========================