课程名称︰微积分乙下 课程性质︰共同必修 课程教师︰陈其诚 开课学院：管理学院 开课系所︰管院各系、地理系、经济系 考试日期（年月日）︰2007.06.05 考试时限（分钟）：110分钟 是否需发放奖励金：是 试题 : Write down your answers on the answer sheet. You should include all the necessary calculations and reasoning. (１) Calculate the following multiple intergrals: 1 1 1 1 (ａ) ∫ ∫ (x + y) dxdy (ｂ) ∫ ∫ e^(-x^2) dxdy 0 0 0 y (7 points) (8 points) (２) Calculate the following double intergrals ∫ ∫f(x, y)dA (7 points each): R (ａ) f(x, y) = xy , R is the triangle with vertices (-π, 0), (π, 0), (π,π/2) (ｂ) f(x, y) = xe^x , R is the triangle bounded by x + y = 4, x = 0, y = 0 y (ｃ) f(x, y) = ──── , 1 + x^2 R is the region bounded by y = 0, y = √x , x = 4 (ｄ) f(x, y) = √(16 - x^2 - y^2) , R is the region bounded by x^2 + y^2 = 16 (ｅ) f(x, y) = 2 , R is the region bounded by 1 ≦ x + y ≦ 2 , 3 ≦ x - 2y ≦ 4. (３) Calculate the triple intergral (10 points) 1 √(1 - x^2) √(1 - x^2 - y^2) ∫ ∫ ∫ √(x^2 + y^2 + z^2) dzdydx -1 -√(1 - x^2) -√(1 - x^2 - y^2) (４) Calculate the triple intergral (10 points) ∫∫∫ x + y + z dV Q where Q is the solid region bounded by 0 ≦ x + y + z ≦ 1 , 0 ≦ y + z ≦ 1 , 0 ≦ z ≦ 1 . (５) Let f(x, y) = xy and let D = {(x, y)│x^2 + y^2 ≦ 1}. (25 points) (ａ) Find critical points of f(x, y) inside the region D. And determine if every critical point is a local extrema. (ｂ) Use the Lagrange multiplier method to determine the maximum / minimum of f(x, y) on the curve x^2 + y^2 = 1 . (ｃ) Find the maximum / minimum of f(x, y) on D. (６) Suppose that f(x, y, z) is a continuous function, And assume that f(0, 0, 0) = 3. For each a, let a √(a^2 - x^2) √(a^2 - x^2 - y^2) V(a) = ∫ ∫ ∫ f(x, y, z) dzdydx -a -√(a^2 - x^2) -√(a^2 - x^2 - y^2) Explain why V(a) lim ──── = 4π (5 points) a→0 a^3 --

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