微積分 XY平面 和函數Z(X,Y) 求質心 20點 急!!

By symmetry, xbar=ybar=0.
V=∫[-1~1]∫[-1~1] (4-x^2-y^2) dy dx
=4∫[0~1]∫[0~1] (4-x^2-y^2) dy dx
=4∫[0~1] (4-x^2- 1/3) dx
=4(4- 1/3 -1/3)=16- 8/3= 40/3
zbar=(1/V)*∫[-1~1]∫[-1~1]∫[0~4-x^2-y^2] zdz dy dx
=(3/40)*2∫[0~1]∫[0~1] (4-x^2-y^2)^2 dy dx
=(3/20)∫[0~1] [(4-x^2)^2-(2/3)(4-x^2)+ 1/5] dx
=(3/20)∫[0~1] [16-8x^2+x^4-(2/3)(4-x^2)+ 1/5] dx
=(3/20)[ 16- 8/3+ 1/5 -(2/3)(4- 1/3)+ 1/5]
=(3/20)(508/45)=127/75