Question

计算∫∫x²/y²dxdy,其中D是由曲线x=2,y=x,xy=1所围成

Answer 1

∫∫_D x²/y² dxdy

= ∫(x = 1→2) x² dx ∫(y = 1/x→x) 1/y² dy

= ∫(x = 1→2) x² * (- 1/y) |(y = 1/x→x) dx

= ∫(x = 1→2) - x² * (1/x - x) dx

= ∫(x = 1→2) (x³ - x) dx

= (x⁴/4 - x²/2) |(x = 1→2)

= (16/4 - 4/2) - (1/4 - 1/2)

= 9/4

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