The region bounded by the curves y = -2x^2 + 2 and y = -x^2 + 1 is shown below:
![Graph of the region bounded by the curves y = -2x^2 + 2 and y = -x^2 + 1]
To find the volume of the solid, we need to integrate the area of each cross section along the x-axis. The area of each cross section is a trapezoid with lower base in the xy-plane, upper base equal to 1/2 the length of the lower base, and height equal to 2 times the length of the lower base.
Let's first find the x-coordinates of the intersection points of the two curves:
-2x^2 + 2 = -x^2 + 1
x^2 = 1
x = ±1
The intersection points are (-1, 2) and (1, 1).
Let's now find the equation of the line that represents the upper base of each trapezoid. The length of the lower base is the difference between the y-coordinates of the two curves:
Length of the lower base = (-x^2 + 1) - (-2x^2 + 2) = x^2 - 1
The length of the upper base is half the length of the lower base:
Length of the upper base = 1/2(x^2 - 1)
The height of the trapezoid is twice the length of the lower base:
Height = 2(x^2 - 1)
The equation of the line that represents the upper base of each trapezoid is:
y = mx + b
where m is the slope and b is the y-intercept. The slope of the line is:
m = (1/2(x^2 - 1) - (-2x^2 + 2)) / (1 - (-1)) = 3x^2/2 - 1/2
The y-intercept of the line is:
b = -2x^2 + 2
Therefore, the equation of the line is:
y = (3x^2/2 - 1/2)x + (-2x^2 + 2)
Simplifying, we get:
y = (3/2)x^3 - (5/2)x + 2