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.
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.