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Find the volume of the solid generated by revolving the region bounded by the graphs of y = x^2 and y = 4x − x about the line y = 6. We should calculate it according to this integral ∫[(6-x^2)^2-(6-(4x-x^2))^2] dx. Why do we integrate it for x-axis not y-axis?

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It is simply that when the line is shifted parallel to x-axis ( y = 6 = constant) sliced thickness radial shell components fall parallel to y-axis, so rotating around x-axis is right. If the shift is parallel to y- axis, then naturally we will choose to rotate around y-axis.

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y= 4x- x? Do you mean y= 4- x? Since you don't put in the limits of integration its hard to tell exactly what you mean. In any case, integrating with respect to x is [b]a[/a] method of finding a volume of integration-the "method of disks" or the "method of washers". Since the x-axis is parallel to the line y= 6, integrating along the x-axis,and rotating around the line y= 6 gives a disk at each specific value of x, or radius 6- y(x) and "thickness" dx. If you have two y-values, then taking away the inner disk gives a "washer".

But it is completely possible to integrate along the y-axis using a given value of x rotates around the line y= 6 to give the radius of a circle so that we have a "cylinder" with x(y) as radius, the difference in the y-values as height and dy as thickness.

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