Question

三重积分难题被积函数为X^2+Y^2,积分区域为Y^2=2Z,X=0绕0Z轴旋转一周而成的曲面与两平面Z=2、Z=8所围

Answer 1

用柱坐标解.

x=r·cos θ;y=r·sin θ;则被积函数X^2+Y^2=r^2;

=∫(从2到8)dz ∫(从0到2π)dθ ∫(从0到√(2Z)) r·r^2 dr

=2π/4∫(从2到8)dz ·r^4|(从0到√(2Z))

=π/2∫(从2到8) (16z^4-0) dz

=8π∫(从2到8)z^4 dz

=(8π/5) z^5|(从2到8)

=(8π/5)(8^5-2^5)

=261888π/5