Question

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.

Answer 1

The base is enclosed by the two parabolas, each with roots at (-1,0) and (1,0)

The lower base B lies between the curves, with length 2(1-x^2)-(1-x^2) = 1-x^2

Using symmetry, then the volume is

2∫[0,1] (B+b)/2 * h dx
= 2∫[0,1] ((1-x^2) + 1/2 (1-x^2))/2 * 2(1-x^2) dx
= 2∫[0,1] 3/2 (1-x^2)^2 dx
= 3∫[0,1] (1-x^2)^2 dx
= 8/5

Answer 2

thanks