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.

why switch names?

To find the volume of the solid, we need to integrate over the region bounded by the curves.

First, let's find the x-coordinates of the intersection points of the two curves. We set the two equations equal to each other and solve for x:

-2x^2 + 2 = -x^2 + 1

Simplifying the equation, we get:

x^2 = 1

Taking the square root of both sides, we find:

x = ±1

So the two curves intersect at x = -1 and x = 1.

Next, let's determine the limits of integration. Since the x-coordinates of the intersection points are -1 and 1, the limits of integration for x should be -1 to 1.

Now let's calculate the area of the cross-section at a given value of x. The lower base of the trapezoid is formed by the curve y = -2x^2 + 2, and the upper base is equal to half the length of the lower base, i.e., (1/2)(2x^2 - 2). The height of the trapezoid is twice the length of the lower base, i.e., 2(2x^2 - 2).

The formula for the area of a trapezoid is given by:

Area = (1/2)(lower base + upper base) * height

Plugging in the values, we get:

Area = (1/2)(-2x^2 + 2 + (1/2)(2x^2 - 2)) * 2(2x^2 - 2)

Simplifying the expression, we have:

Area = (1/2)(-2x^2 + 2 + x^2 - 1) * 4x^2 - 4

Area = (1/2)(-x^2 + 1) * 4x^2 - 4

Finally, we integrate the expression over the limits of integration from -1 to 1 to find the volume:

Volume = ∫[from -1 to 1] [(1/2)(-x^2 + 1) * (4x^2 - 4)] dx

Solving this integral will give us the volume of the solid.