題目 6. (10 points) Evaluate ∫022∫y1−y2ex2+y2 dx dy.\int_{0}^{\frac{\sqrt{2}}{2}} \int_{y}^{\sqrt{1-y^2}} e^{x^2+y^2}\,dx\,dy.∫022∫y1−y2ex2+y2dxdy. 解答 解法一 積分區域為單位圓內、0≤θ≤π/40\le\theta\le\pi/40≤θ≤π/4 的扇形。故 ∬Rex2+y2 dA=∫0π/4∫01er2r dr dθ=π(e−1)8.\iint_R e^{x^2+y^2}\,\mathrm dA =\int_0^{\pi/4}\int_0^1e^{r^2}r\,\mathrm dr\,\mathrm d\theta =\boxed{\frac{\pi(e-1)}8}.∬Rex2+y2dA=∫0π/4∫01er2rdrdθ=8π(e−1).
