The region in the first quadrant enclosed between the graph of y=ax-x^2 and the x-axis generates the same volume whether it is revolved about the x-axis or the y-axis. find the value of a

which means the parabola intersects the x-axis at and . Assume .

Then the volume of the solid generated by revolving the region about the x-axis is



Revolving about the y-axis, the volume would be



The volumes are the same independent of which axis is taken as the axis of revolution, so



We're assuming , so we can safely divide both sides by to get .