{ z = - √(x² + y²)
{ z = - 1
- 1 = - √(x² + y²)
x² + y² = 1 --> r = 1
切片法:
∫∫∫ z dV
= ∫(- 1→0) z dz ∫∫Dz dxdy
= ∫(- 1→0) z * πz² dz
= ∫(- 1→0) πz³ dz
= (π/4)(z⁴):[- 1→0]
= π/4 * 0 - π/4 * (- 1)⁴
= - π/4
投影法:
∫∫∫ z dV
= ∫(0→2π) ∫(0→1) ∫(- 1→r) rz dzdrdθ
= 2π∫(0→1) r * z²/2:[- 1→r] dr
= 2π∫(0→1) r/2 * (r² - 1) dr
= π∫(0→1) (r³ - r) dr
= π(r⁴/4 - r²/2):[0→1]
= - π/4