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The Second Midterm Examination of Calculus

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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.

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