Calculus (Solid of Revolution)
posted by Casablanca .
The region R is bounded by the xaxis, x = 2, and y = x^2. What is the the volume of the solid formed by revolving R about the line x = 2?

y=x^2 and x=2 intersect at (2,4)
the radius of the rotated region = 2  x
= 2  √y
V = π∫ (2√y)^2 dy from 0 to 4
= π∫ (4  4√y + y) dy from 0 to 4
= π[ 4y  (8/3)y^(3/2) + (1/2)y^2 ] from 0 to 4
= π (16  (8/3)(8) + 8  0)
= 8π/3
check my arithmetic and thinking. 
y=x^2 intersects x=2 at (2,4)
using shells,
v = ∫2πrh dx [0,2]
where
r = 2x
h = y
v = 2π∫(2x)(x^2)dx [0,2]
= 2π∫2x^2  x^3 dx [0,2]
= 2π(2/3 x^3  1/4 x^4)[0,2]
= 2π(16/3  4)
= 2π(4/3)
= 8π/3
using discs,
v = ∫πr^2 dy [0,4]
= π∫(2√y)^2 dy [0,4]
= π∫(4  4√y + y)dy [0,4]
= π(4x  8/3y√y + 1/2 y^2)[0,4]
= π(16  64/3 + 8)
= 8π/3
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