題目 6. (10%)(10\%)(10%) Evaluate ∬Dln(x2+y2)x2+y2 dx dy,\iint_D \frac{\ln(x^2+y^2)}{\sqrt{x^2+y^2}}\,dx\,dy,∬Dx2+y2ln(x2+y2)dxdy, where D={(x,y):1≤x2+y2≤e}D=\{(x,y):1\le x^2+y^2\le e\}D={(x,y):1≤x2+y2≤e}. 解答 解法一 答題過程 展開 極坐標範圍為 1≤r≤e,0≤θ≤2π1\le r\le\sqrt e,0\le\theta\le2\pi1≤r≤e,0≤θ≤2π，故 ∫02π∫1elnr2rr dr dθ=2π(2−e).\int_0^{2\pi}\int_1^{\sqrt e}\frac{\ln r^2}{r}r\,\mathrm{d}r\,\mathrm{d}\theta=\boxed{2\pi(2-\sqrt e)}.∫02π∫1erlnr2rdrdθ=2π(2−e).
