3. The base of the solid is the region bounded by the parabola y²= 8x and the line x=8. Find the volume of the solid if every plane section perpendicular to the axis of the base is a square.

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Answer 1

When x=8, y=±8, so the side of the square cross-section is 16.

The parabola is arranged sideways with vertex at (0,0) and arms extended to the right. The area of the square base is 16²=256 square units. If we take any point along the x-axis between x=0 and x=8 the length of the side of the square is 2√(8x), so the area of the square cross-section is 32x. (When x=8 this is 256 as we calculated earlier.) The thickness of the square is dx so we have 32xdx as the volume of an infinitesimally thin square slice anywhere between x=0 and 8, So the integral is:

₀∫⁸32xdx=[16x²]₀⁸=16×64=1024 cubic units.