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计算二重积分∬(6x+3y)dσ在x+y=14与两坐标轴围成区域D的值。
主要内容：
介绍通过两种不同的积分次序，计算二重积分∬(6x+3y)dσ在区域D上的值，其中D是直线x+y=14与两坐标轴围成的区域。
思路一：先对dy积分，再对dx积分
∬(6x+3y)dσ
=∫[0,14]dx∫[0,(14-x)](6x+3y)dy
=∫[0,14]dx(6xy+3/2*y^2)[0,(14-x)]
=∫[0,14][6x(14-x)+3/2*(14-x)^2]dx
=3/2∫[0,14](196+28x-3x^2)dx
=3/2(196x+14*x^2-x^3)[0,14]
=3/2[196*14+14*14^2-14^3]
=4116。
思路二：先对dx积分，再对dy积分
∬(6x+3y)dσ
=∫[0,14]dy∫[0,(14-y)](6x+3y)dx
=∫[0,14]dy(3x^2+3yx)[0,(14-y)]
=∫[0,14][3(14-y)^2+3y*(14-y)]dy
=42∫[0,14](-y+14)dy
=42(-1/2*y^2+14y)[0,14]
=42(1/2*14^2+14*14)
=4116。
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