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प्रश्न
Evaluate `int int xy(x-1)dx dy` over the region bounded by 𝒙𝒚 = 𝟒,𝒚= 𝟎,𝒙 =𝟏 and 𝒙 = 𝟒
उत्तर
Let I = `int int xy(x-1)dx dy`
Rectangular hyperbola : 𝒙𝒚=𝟒 Lines : 𝒙=𝟏 ,𝒙=𝟒 ,𝒚=𝟎
Intersection of line 𝒙 = 𝟏 and 𝒙𝒚 = 𝟒 is (1,4).
Intersection of line 𝒙 = 𝟒 and 𝒙𝒚 = 𝟒 is (4,1)
`therefore 0<=y<=x/4`
`1<=x<=4`
`therefore "I" =int_1^4 int_0^(x/4)(x^2y-xy)dy dx`
`=int_1^4[y^2/2x^2-(y^2x)/2]_0^(x/4)dx`
`=int_1^4(8-8/x)dx`
`=[8x-8logx]_1^4`
`therefore "I"=8(3-2log2)`
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