sketch the region R bounded by the graphs of the equations, and find the volume of the solid generated if R is revolved about the x axis when y=x^2 y=4-x^2

Answer 1

The curves intersect where x = ±√2.

By symmetry, using discs,

v = 2∫[0,√2] π(R^2-r^2) dx
where R = 4-x^2, and r = x^2

v = 2π∫[0,√2] ((4-x^2)^2 - (x^2)^2) dx
= 2π∫[0,√2] (16 - 8x^2 + x^4 - x^4) dx
= 2π∫[0,√2] (16 - 8x^2) dx
= 2π(16x - 8/3 x^3)[0,√2]
= 2π(16√2 - 16/3 √2)
= 64π√2/3