Calculus: Early Transcendentals, Chapter 6, 6.3, Section 6.3, Problem 34

y= x^3-x+1
y = -x^4+4x-1
The graph of the two equations are:

(Green curve graph of y=x^3-x+1 . Blue curve graph of y = -x^4+4x-1 .)
Base on the graph, the curve curves intersect at x~~0.421 and x~~1.23 .
To solve for the volume of the solid formed when the bounded region is rotated about the y-axis, apply method of cylinder. Its formula is:
V= int _a^b 2pi *r * h*dx
To determine the radius and height of the cylinder, refer to the figure below. Base on it, its radius and height are:
r = x
h = y_(upper) - y_(lower)
h=(-x^4+4x-1) - (x^3-x+1) = -x^4-x^3+5x - 2
Plugging them to the formula of volume yields:
V=int_0.42^1.23 2pi *x *(-x^4-x^3+5x-2)dx
V=2pi int_0.42^1.23 (-x^5-x^4+5x^2-2x)dx
V= 2pi (-x^6/6 - x^5/5+(5x^3)/3-x^2)|_0.42^1.23
V=2pi [ (-1.23^6/6-1.23^5/5+(5*1.23^3)/3-1.23^2)-(-0.42^6/6-0.42^5/5+(5*0.42^3)/3-0.42^2)]
V=2pi*0.5048
V=3.1717
Therefore, the volume of the solid formed is 3.1717 cubic units.