Volumes: Cross-Sectionals

Question

Find the volumes of the solids whose bases are bounded by the circle x² + y² = 4, with the indicated cross sections taken perpendicular to the x-axis.

(a)semi-circles

(b) isosceles right triangles

Answer 1

a shortcut: the semicircles make a hemisphere, so the volume is 2/3πr³ = 2/3π2³ = 16/3 π

Since the isosceles triangles would each have, where half the cross-section has length c, area c² because the hypotenuse is 2c, so legs are each √2 c, and A = 1/2(√2 c)2 = c², while the semicircles have cross-section π/2 c², 2/π (16/3 π) = 32/3

The long way: each semicircle at x will have radius ⋅√(4-x²),so ∫₂² π/2 (4 - x²) dx =

π/2 (4x - x³/3)|_-2² =

π/2((4*2-2³/3)-(4*-2 - (-2)³/3)) =

π/2(16 - 16/3) =

16/3π

each triangle will have area ⋅4-x²,so ∫₂² (4 - x²) dx =

(4x - x³/3)|_-2² =

((4*2-2³/3)-(4*-2 - (-2)³/3)) =

(16 - 16/3) =

32/3