[Risolto] Rappresenta la regione limitata dal grafico della funzione y=Inx, dalla retta y=1 e dalla funzione y=1-Inx . Calcolane l'area .

RAPPRESENTAZIONI
https://www.wolframalpha.com/input/?i=y+%3D+1+-+ln%28x%29
https://www.wolframalpha.com/input/?i=y+%3D+ln%28x%29
https://www.wolframalpha.com/input/?i=%5By%3Dln%28x%29%2Cy%3D1-ln%28x%29%2Cy%3D1%5Dx%3D0to3
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INTERSEZIONI
* (y = 1 - ln(x)) & (y = 1) ≡ A(1, 1)
* (y = 1 - ln(x)) & (y = ln(x)) ≡ B(√e, 1/2)
* (y = ln(x)) & (y = 1) ≡ C(e, 1)
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AREA
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* ∫ ln(x)*dx = x*(ln(x) - 1) + c
* ∫ [x = a, b] ln(x)*dx = b*(ln(b) - 1) - a*(ln(a) - 1)
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* A = ∫ [x = 1, √e] (1 - (1 - ln(x)))*dx + ∫ [x = √e, e] (1 - ln(x))*dx =
= ∫ [x = 1, √e] ln(x)*dx + ∫ [x = √e, e] dx - ∫ [x = √e, e] ln(x)*dx =
= ∫ [x = 1, √e] ln(x)*dx + (e - √e) - ∫ [x = √e, e] ln(x)*dx =
= (√e)*(ln(√e) - 1) - 1*(ln(1) - 1) + (e - √e) - (e*(ln(e) - 1) - (√e)*(ln(√e) - 1)) =
= (√e - 1)^2 ~= 0.42