MATH 141, FALL 2008, MIDTERM 1 - REVIEW

1) Find the value of the following limit, if it exists:

lim_{x→∞} (1 + \frac{1}{x^{2}} )^{x} .

If the limit does not exist, explain why.

2) Let f(x) = 1/(1 − x^{2})^{1/4}, and let R be the region between the graph of f and the x axis on [0, 1/2]. Find the volume V of the solid obtained by revolving R about the x axis.

3) Find the length of a curve given parametrically by

x = f(t) = e^{t} sin(t)

and

y = g(t) = e^{t} cos(t),

for 0 ≤ t ≤ π/2.

4) Evaluate the integral:

∫ e^{x}(1 − e^{2x})^{−0.5} dx

5) A swimming pool has the shape of a right circular culinder with radius 10 feet and depth 8 feet. If the pool contains 5 feet of water, what is the work required to pump all the water to the top of the pool?