The first quadrant region enclosed by the x-axis and the graph of y = ax - x^2 traces out a solid of the same volume whether it is rotated about the x-axis or the y-axis. What is the value of a?
2026-04-02 20:12:53.1775160773
Revolving an unknown equation around the x and y axes
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Hint: $$\pi\int_0^a (ax-x^2)^2dx= 2\pi\int_0^ax(ax-x^2)dx$$
The upper bound of integration of both integrals is $a$ because that is where the function will cross the x-axis (assuming $a>0$).