Math
posted by jake on .
Find the volume of the solid generated by revolving the following region about the given axis
The region in the first quadrant bounded above by the curve y=x^2, below by the xaxis and on the right by the line x=1, about the line x=4

I'd suggest using shells for this one:
v = ∫[0,1] 2πrh dx
where r = x+4 and h = y = x^2
v = 2π∫[0,1](x+4)*x^2 dx
= 2π∫[0,1] x^3 + 4x^2 dx
= 2π (1/4 x^4 + 4/3 x^3) [0,1]
= 2π (1/4 + 4/3)
= 19π/6
It can be done with discs, but you have to make them washers:
v = ∫[0,1] π(R^2r^2) dy
where R = 5, r=4+x = 4+√y
v = π∫[0,1] (25  (4+√y)^2) dy
= π (9y  16/3 y^3/2  1/2 y^2) [0,1]
= π (9  16/3  1/2)
= 19π/6