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