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Find the area of the set

Source: 2009 József Wildt International Mathematical Competition

April 15, 2020
algebra

Problem Statement

Find the area of the set A={(x,y) ∣ 1≤x≤e, 0≤y≤f(x)}A = \{(x, y)\ |\ 1 \leq x \leq e,\ 0 \leq y \leq f (x)\}, where \begin{tabular}{ c| c c c c |} &1 & 1& 1 & 1\\ f(x)f(x)=& ln⁡x\ln x & 2ln⁡x\ln x & 3ln⁡x\ln x & 4ln⁡x\ln x \\ &(ln⁡x)2{(\ln x)}^2 & 4(ln⁡x)24{(\ln x)}^2 & 9(ln⁡x)29{(\ln x)}^2 & 16(ln⁡x)216{(\ln x)}^2\\ &(ln⁡x)3{(\ln x)}^3 & 8(ln⁡x)38{(\ln x)}^3 &27(ln⁡x)3 27{(\ln x)}^3 &64(ln⁡x)3 64{(\ln x)}^3 \end{tabular}
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