https://docs.google.com/document/d/1Mmp2...
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Since we are integrating over rectangular region, we can change order of integration:
∫ [1 to 2] ∫ [1 to 2] y/(x+y²) dx dy
= ∫ [1 to 2] ∫ [1 to 2] y/(x+y²) dy dx
= ∫ [1 to 2] (1/2 ln(x+y²) | [1 to 2] ) dx
= 1/2 ∫ [1 to 2] (ln(x+4) − ln(x+1)) dx
= 1/2 ((x+4) ln(x+4) − (x+1) ln(x+1)) | [1 to 2]
= 1/2 [ (6 ln(6) − 3 ln(3)) − (5 ln(5) − 2 ln(2)) ]
= 1/2 (6 ln(2) + 6 ln(3) − 3 ln(3) − 5 ln(5) + 2 ln(2))
= 1/2 (8 ln(2) + 3 ln(3) − 5 ln(5))
= 1/2 ln(6912/3125)
≈ 0.396912374
∫ [1 to 2] ∫ [1 to 2] y/(x+y²) dx dy
= ∫ [1 to 2] ∫ [1 to 2] y/(x+y²) dy dx
= ∫ [1 to 2] (1/2 ln(x+y²) | [1 to 2] ) dx
= 1/2 ∫ [1 to 2] (ln(x+4) − ln(x+1)) dx
= 1/2 ((x+4) ln(x+4) − (x+1) ln(x+1)) | [1 to 2]
= 1/2 [ (6 ln(6) − 3 ln(3)) − (5 ln(5) − 2 ln(2)) ]
= 1/2 (6 ln(2) + 6 ln(3) − 3 ln(3) − 5 ln(5) + 2 ln(2))
= 1/2 (8 ln(2) + 3 ln(3) − 5 ln(5))
= 1/2 ln(6912/3125)
≈ 0.396912374