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Question
Evaluate `int int xy(x-1)dx dy` over the region bounded by ๐๐ = ๐,๐= ๐,๐ =๐ and ๐ = ๐
Solution
Let I = `int int xy(x-1)dx dy`
Rectangular hyperbola : ๐๐=๐ Lines : ๐=๐ ,๐=๐ ,๐=๐
Intersection of line ๐ = ๐ and ๐๐ = ๐ is (1,4).
Intersection of line ๐ = ๐ and ๐๐ = ๐ 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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