1. Volume by cross-sections (possibly misleading)

The basis of a solid is the circle . Every cross-section perpendicular to the x axis is a rectangle whose height is twice the distance from the origin of the circle to the plane of the cross-section. Find the volume of the solid.

Note: I don't know what the problem means by "origin of the circle". I assumed it means the center of the circle, (8,0), but I think it could mean instead the origin (0,0).

This is my reasoning: the distance mentioned is , so the height of a cross-section is , assuming . The length of a cross-section is , so the cross-sectional area is . Integrate this from x = 0 to x = 8, multiply by 2 to cover the other half of the solid and you find . But the answer is supposed to be or something like that.

Is this right? Is the problem misleading? Any help would be greatly appreciated.

2. $(x-8)^2 + y^2 = 8^2$

horizontal distance from the origin (0,0) to a representative cross-section is x.

area of a representative cross-section ...

$A = 2x \cdot 2y = 4xy = 4x\sqrt{64-(x-8)^2}$

$V = \int_0^{16} 4x\sqrt{64-(x-8)^2} \, dx = 1024\pi$