[試題] 103-2 洪立昌 微積分甲下 第六次小考

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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

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