Use the cylindrical shell method to compute the volume Be su

Solution

For the cylindrical shell method, the element you need to rotate about the x-axis to generate the same geometrical volume is the horizontal element of thickness dy extending from the point (x, y) on the curve to the upper limit of x (which I assume from the information provided is x = 1), so that the elemental volume of each cylinder is dV = 2.p.y.(1 - x).dy, where y is the radius of the cylinder and (1 - x) its height. Then we have V = 2.p ? y.(1 - x).dy = 2.p ? y.[1 - y^(1/3)].dy = 2.p ? [y - y^(4/3)].dy . = 2.p.[(1/2).y² - (3/7).y^(7/3)] with limits y = 0 to 1 . = p.[1 - (6/7)] = p/7. I suspect that you took the height of the cylinder as x rather than (1 - x), which leaves you with only the second term in the integral, which is indeed 6.p/7.