[試題] 103-2 洪立昌 微積分甲下 第五次小考

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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5日
考試時限（分鐘）：60分鐘

試題 :

Instructions : (120pts) Show all details as possible as you can.

   (10pts)
             3 2
1. Evaluate ∫∫ x^2 y dydx.
             0 1

   (15pts)
                                                3 2
2. Sketch the region and evaluate the integral ∫∫     e^(x/(y+1)) dydx.
                                                0 √x+1

   (20pts)
                                   a  a
3.Evaluate the iterated integrals ∫ ∫ sin(y^2) dydx, a > 0 and
   1  1                            0  x
  ∫ ∫   sin(t^3) dtdx.
   0  √x


   (20pts)

4.Evaluate ∫∫ xy dA , where D is bounded by line y = x-1 and the parabola
              D
  y^2 = 2x + 6.


  (10pts)
            1                                1                1
5.Evaluate ∫ x-1/lnx dx by rewriting it as ∫(x-1)/lnx dx = ∫ x^1-x^0/lnx dx.
            0                                0                0

  (30pts)

6. Suppose that f' is continuous and the integral converges, then
    ∞
   ∫ (f(ax)-f(bx))/x dx = [f(∞)-f(0)]ln(a/b) , where a,b > 0 are constants.
    0

  (i) Let D = {(x,y) ∈ R^2 : x≧0 , a≦y≦b}. Show the Frullani's integral
      above by evaluating the integral

                             I = ∫∫ -f'(xy)dxdy
                                    D
                                  b  ∞                      ∞ b
      in two differents ways I = ∫(∫ -f'(xy)dx)dy and I = ∫(∫ -f'(xy)dy)dx.
                                  a  0                       0  a

                    1
  (ii) Evaluate    ∫ (sin^-1(17x) - sin^-1(23x)) / x  dx.
                    0

  (15pts)

7.Suppose that f(x) is continuous and f(x) > 0 on [a,b]. Prove that
                b         b
              (∫f(x)dx)(∫ 1/f(x) dx) ≧ (b-a)^2
                a         a

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