find the volume of the solid of revolution formed by revolving the region bounde

Question

find the volume of the solid of revolution formed by revolving the region bounded by y=x-x^2 and the x-axis about the y-axis

I am supposed to use the shell method to solve this but I am not quite sure how to work it.

Explanation / Answer

y = + x - x² y = - x - x = + x - x² x² - x = 0 x (x - 1) = 0 x = 0, x = 1 o we integrate from x = 0 to x = 1 On this interval 1 + x - x² > 1- 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 = (1+ x - x²) - (1 - x) = x - x² V = 2p ?0² r h dx V = 2p ?0² x (1x - x²) dx V = 2p ?0² (1x² - x³) dx Solving, we get V = 4p/3

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