99年 - 99 國立政治大學轉學生招生考試_應用數學系：微積分(二)#122370

(a) (5%) ∫ dx 

(b) (5%) ∫ dx

(a) (10%) For a certain real number a, the following integral
  converges. Determine a and evaluate the integral.

(b) (10%) For what values of the constants b and cwill the following limit exist and be equal to 1? dx .

3. (15%) If dis an arbitrary real number, let Sn(d) =. Determine the following limit: .

4. (10%) Let f(x) be a polynomial of degree m. Show that ∫e^xf(x)dx = e^x(f(x) - f'(x) + f"(x) - ... + (-1)^mf^(m)(x))

5. (15%) Evaluate the following integral
  
along the curve C = {(x,y,z) ∈ R³ | x² + y² = 4, z = -3}, oriented counterclockwise as seen by a person standing at the origin and with respect to the right-handed Cartesian coordinates, where
F = yi + xz³j – zy³k,
 r' = dr/ds is the unit tangent vector, and *sis the arc length of C

7. (15%) Compute the area of the hemisphere of radius t>0 and centered at the origin given by S = {(x,y,z) ∈ R³| x² + y² + z² = t², z≥0}