Question

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis.
x+y=4,x=5−(y−1) 2 ;

Answer 1

The curves intersect at (4,0) and (1,3).

You don't specify the axis, so, assuming the x-axis,

using shells, the volume is
v = ∫[0,3] 2πrh dy
where r = y and h = (5-(y-1)^2)-(4-y)
v = 2π∫[0,3] 3y^2 - y^3 dy
= 27π/2

Using washers, we have to break the parabola into its two branches:
y = 1+√(5-x) and y = 1-√(5-x)

v =
∫[1,4] π(R^2-r^2) dx
where R=1+√(5-x) and r = 4-x
+∫[4,5] π(R^2-r^2) dx
where R = 1+√(5-x) and r = 1-√(5-x)

So,
v = ∫[1,4] π((1+√(5-x))^2-(4-x)^2) dx
+ ∫[4,5] π((1+√(5-x))^2-(1-√(5-x))^2) dx
= 65π/6 + 8π/3
= 27π/2

If you meant to revolve around the x-axis, just swap some stuff around.