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A solid has as its base the region bounded by the curves y = –2x^2 + 2 and y =-x^2+ 1. Find the volume of the solid if every cross section of a plane perpendicular to the x-axis 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.

A solid has as its base the region bounded by the curves y = –2x^2 + 2 and y =-x^2+ 1. Find-example-1

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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
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