Sketch the graphs of f(x) = x^2 and g(x) = 2x. Find a number a between 0 and 2 so that when the region between the graphs of f and g for 0 <= x <= a is revolved about the y-axis, the volume of the solid produced is 7pia3/12 . Please show all work

Answer 1

v = ∫[0,a] π(R^2-r^2) dx
= π∫[0,a] (2x)^2 - (x^2)^2 dx
= πa^3(4/3-a^2/5)

so, we want πa^3(4/3-a^2/5) = 7/12 π a^3
4/3 - a^2/5 = 7/12
80 - 12a^2 = 35
12a^2 = 55
a = √(55/12)

Answer 2

oops: √45/12 = √(15/2)