Question

Use the shell method to find the volume of the solid generated by revolving the plane region about the indicated line.

y=4x-x^2, y=0, about the line x=5

Answer 1

v = ∫[0,4] 2πrh dx
where r=5-x and h=y
= 2π∫[0,4] (5-x)(4x-x^2) dx
= 2π(x^4 - 3x^3 + 10x^2) [0,4]
= 64π

check using discs:

v = ∫[0,4] π(R^2-r^2) dy
where R=5-(2-√(4-y)) and r=5-(2+√(4-y))
= π∫[0,4] (3+√(4-y))^2-(3-√(4-y))^2
= π∫[0,4] (9+6√(4-y)+(4-y))-(9-6√(4-y)+(4-y)) dy
= π∫[0,4] 12√(4-y) dy
= 4π (-2 (4-y)^(3/2)) [0,4]
= 64π