∫∫(x*x+4y*y+9)dxdy,其中D={9x,y0|x*x+y*y

Question

∫∫(x*x+4y*y+9)dxdy,其中D={9x,y0|x*x+y*y

Answer 1

∫∫(x*x+4y*y+9)dxdy
=∫∫(x²+4y²)dxdy+9∫∫1dxdy
前一个用极坐标
=∫∫(r²+3r²sin²θ)rdrdθ+9*π*2²
=∫∫(r²+3r²sin²θ)rdrdθ+36π
=∫ [0--->2π]dθ∫[0--->2] r³(1+3sin²θ)dr+36π
=∫ [0--->2π](1+3sin²θ)dθ∫[0--->2] r³dr+36π
=(1/4)∫ [0--->2π](1+3sin²θ)r⁴ |[0--->2]dθ+36π
=4∫ [0--->2π](1+3sin²θ)dθ+36π
=4∫ [0--->2π](1+(3/2)(1-cos2θ))dθ+36π
=2∫ [0--->2π](5-3cos2θ))dθ+36π
=2(5θ-(3/2)sin2θ) |[0--->2π]+36π
=20π+36π
=56π