Re: [積分] 請教一題求體積
※ 引述《yummy100 (LING)》之銘言：
: 拜託~幫忙解答~
: 求 ∫∫∫(x^2 + y^2 + z^2) 所圍體積? R:{ x+y+z=a , x=0, y=0, z=0}
: R
: Ans: a^5 /20
a a-z a-z-y
∫ ∫ ∫ (x^2 + y^2 + z^2) dxdydz
0 0 0
a a-z
=∫ ∫ (1/3)*(a-z-y)^3 + (y^2 +z^2)(a-z-y) dydz
0 0
a
=∫ (1/2)*(z^2)*(a-z)^2 + (1/6)*(a-z)^4 dz
0
= (1/2)*[(a^5)/3 - (a^5)/2 + (a^5)/5] - (-1/30)a^5
= (1/60 + 1/30)a^5
= (1/20)a^5
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