The region bounded by y=-x^2+14-45 and y=0 is rotated about the y-axis, find the volume.

Answer 1

The region goes from x=5 to x=9, So,

using shells,

v = ∫[5,9] 2πrh dx
where r = x and h=y
v = 2π∫[5,9] x(-x^2+14x-45) dx
= 448/3 π

Using discs (washers), things get a bit more complicated, because there are two branches to the parabola.

y = 4-(x-7)^2
x = 7±√(4-y)

v = ∫[0,4] π(R^2-r^2) dy
where R = 7+√(4-y) and r = 7-√(4-y)
v = π∫[0,4] (7+√(4-y))^2 - (7-√(4-y))^2) dy
= 448/3 π