題目 Problem 10 (10%). Let RRR be the region bounded by the graph of xy=1xy=1xy=1, xy=4xy=4xy=4, x=2x=2x=2, and x=3x=3x=3. Compute the integral ∬Re−xy1+x2 dA.\iint_R \frac{e^{-xy}}{1+x^2}\,dA.∬R1+x2e−xydA. 解答 解法一 令 u=xyu=xyu=xy、v=xv=xv=x，則 dA=1x du dv\mathrm{d}A=\frac1x\,\mathrm{d}u\,\mathrm{d}vdA=x1dudv，範圍為 1≤u≤41\le u\le41≤u≤4、2≤v≤32\le v\le32≤v≤3。故 ∬Re−xy1+x2 dA=(e−1−e−4)ln324.\iint_R\frac{e^{-xy}}{1+x^2}\,\mathrm{d}A =\boxed{(e^{-1}-e^{-4})\ln\frac{3\sqrt2}{4}}.∬R1+x2e−xydA=(e−1−e−4)ln432.
