92ç,ç‚Bø ‚5
1. The curve y2 = x 1 3 − x
2
encloses a bounded plane region. Find the area of this plane region (5%) and the arc length of the boundary curve (5%). Now we revolve this plane region around y-axis to obtain a solid of revolution. Find the volume of this solid (5%).
2. Find the limit lim
x→0
1
x2 − 1 tan2x
. (10%)
3. Find the derivative d dx
xsec−1(2x) − 1
4ln 1 + 4x2
. (10%)
4. Find the antiderivative
Z ex+ 1
e2x− ex+ 2dx. (10%)
5. Evaluate the integral Z π2
0
sinnθ dθ, n ∈ N. (10%)
6. Find the improper integral Z 1
0
1
px(1 − x) dx. (10%)
7. An observer at height H above the north pole of a sphere of radius r. Find the surface area that he can see. (10%)
8. Test the series for convergence or divergence.
(a)
∞
X
n=1
(−1)n
n n+ 1
n2
. (5%)
(b)
∞
X
n=2
(−1)n−1
√n − 1. (5%)
(c)
∞
X
n=1
n
n2+ 3. (5%)
9. Find the Taylor series expansion of the function ex2 about x = 0 and prove that it converges to ex2. (10%)
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