The base of a solid S is the region enclosed by the parabola y = 4 − x2 and the x-axis. Cross-sections perpendicular to the y-axis are squares.

Answer 1

The base is bounded above by the parabola y = 4 - x² and below by the x-axis. The parabola has y-intercept (0,4) and x-intercepts (2,0) and (-2,0).

A typical horizontal slice of the base intersects the parabola at the point (x, 4-x²), 0 ≤ x ≤ 2. the slice has length 2x.

Volume of typical cross section = (2x)(2x)Δy = 4x²Δy = 4(4-y)Δy

Volume of solid = ∫_{(0 to 4)}(16 - 4y)dy = [16y - 2y²]_{(0 to 4)} = 64 - 32 = 32

Answer 2

Volume v = ∫₀⁴[2√(4 - y)]²dy = 4∫(4 - y)dy = - 2(4 - y)²₀⁴ = 32