The Second Midterm Examination of Calculus
2009/12/2 1. (10%) Calculate (i) ∫ b3x3
√1−a4x4dx; (ii) ∫1 0
√x+3 x+1dx.
2. (15%) Calculate the following integrals (i) ∫
sin6xdx (ii)
∫ 2
−2
√ x
x2 + 5√
1 +√
x2 + 5 dx.
3. (10%) Find lim
n→+∞
∑n i=1
sin(iπn)
n by evaluating an appropriate definite integral over the interval [0, 1].
4. (10%) Calculate the area of the region bounded by the curves x = y2 and x− y = 2 by integrating with respect to x.
5. (20%) Find the volume of the solid generated by revolving the region between y = x3 and y = √
x about the x-axis and y-axis.
6. (8%) Find H0(3) given that H(x) = 1x∫x
3 [2t − 3H0(t)]dt, where H0(t) is continuous.
7. (12%) Defined F by F (x) = ∫x 0 [t∫t
1
√u2 + 1 du]dt. Find F0(x), F0(1), F00(x) and F00(1).
8. (15%) Sketch the graph of the function f (x) = x2 − 3 x3 and show all geometric characteristics.