課程名稱︰微積分甲下 課程性質︰土木系大一必修 課程教師︰洪立昌 開課學院：理學院 開課系所︰數學系 考試日期（年月日）︰2015年5月19日 考試時限（分鐘）：60分鐘 試題 : Instructions : (130 points) Show all details as possible as you can. (25pts) (15.4 Double Integrals in Polar Coordinates) (√3)/2 √(1-y^2) 1.Please evaluate ∫ ∫ sin(x^2 + y^2)dxdy by answering the following 0 y/√3 questions in order (i) Find the intersection point(s) of x = √(1-y^2) and x = y/√3 in the first quadrant.(3 %) (ii) Find the angle (less then π/2) between the line x = y/√3 and the x-axis (3 %) (iii) Plot the region of the integral domain; label the intersection point(s) in (i) and the angle in (ii). (3 %) (iv) Describe the integral domain in polar coordinates. (3 %) √3 /2 √(1-y^2) (v) Evaluate ∫ ∫ sin(x^2 + y^2) dxdy using the results in 0 y/√3 (iii) and (iv). (13%) (25pts) (15.6 Surface Area) 2. Find the surface area of the part (denote by A) of the sphere x^2 + y^2 + z^2 = 4 that is above the plane z = 1 by answering the following questions in order : (i) Describe A by z = f(x,y), where x,y ∈ D. Find f(x,y) and D.(8%) (ii) Write down the integral formula of the area of A in rectangular coordinates. (no need to evaluate it here) (5%) (iii) Write down the integral formula of the area of A in polar coordinates. Then find the surface area of A. (12%) (25pts) (15.9 Triple Integrals in Spherical Coordinates) 3. Find the volume of the solid S that lies above the cone z = √(x^2 + y^2) and below the sphere x^2 + y^2 + z^2 = 1 by answering the following questions in order : (i) Describe z = √(x^2 + y^2) in spherical coordinates x = ρsinψcosθ, y = ρsinψsinθ,z = ρcosθ. (4%) δ(x , y , z) (ii) Write down the Jacobian ────── .(no need to proof it.)(5%) δ(ρ,ψ,θ) ps.δ代表偏微分符號 ( ptt打不出來 QQ ) (iii) Find the volume of the solid S using the results in (i) and (ii). (16%) (25pts) (15.10 Change of Variables in Multiple Integrals) 4. Please elaluate ∫∫ cos ((y-x)/(y+x)) dA, where R is the trapezoidal region R with vertices (1,0),(2,0),(0,2),and(0,1) by answering the following question in order: δ(x , y) (i) Let u = y-x , v = y+x and find the Jacobian ─────.(5%) δ(u , v) (ii) Plot the region of the integral domain R in the uv-plane and label all the vertices of it. (5%) (iii) Evaluate ∫∫ cos((y-x)/(y+x))dA using the results in (i) and (ii) (15%) R (30pts) (15.7 Triple Integrals : Problem Plus and Bonus) x y z x 5. Assume that f is continuous.Prove ∫∫∫f(t)dtdzdy = 1/2 ∫ (x-t)^2 f(t)dt 0 0 0 0 (1) by answering the following questions in order: y z (i) Evaluate ∫∫ f(t)dtdz by changing the order of integration.(10%) 0 0 (ii) Using the result in (i), prove (1) again by changing the order of integration.(10%) (iii) If you finish (i) and (ii) , could you use similar arguements to conjecture and evaluate the following quadruple (consisting of four partsof members) intrgral? (10%) s x y z ∫∫∫∫ f(t) dtdzdydx. 0 0 0 0 -- 標題 [問卦] 她跟iPad3有什麽關係? --

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