Description
Problem 1
(10 points)
Find
Z 0
−1
ln(x + 1) dx .
Problem 2
(10 points)
a) Show that the volume enclosed when revolving the curve y = f(x) – where f : [a, b] →
[0,∞) – about the x-axis in three-dimensional x-y-z space is given by
V = π
Z b
a
f
2
(x) dx .
Hint: Think about the cross-sectional areas and the perfect symmetry when revolving the
function around the x-axis. (5 points)
b) Compute the volume of the solid obtained by revolving the graph of of y =
1 + x
2
2
on
[0, 1] about the x-axis. (5 points)
Problem 3
(10 points)
a) Hook’s law states that the force exerted by an ideal spring when extended from its equilibrium position at x = 0 to length x is given by
F(x) = −k x ,
where k is a positive constant characterizing the stiffness of the spring. Compute the work
required to expand the spring from its equilibrium position to length `. (5 points)
b) Show that
Z ∞
1
1
√
1 + x
4
dx
is convergent. (5 points)
Hint: There is no elementary way to evaluate this integral. However, to only test convergence, you can bound the integrand by a simpler function and use the following fact without proof: Let f : [a,∞) → R be a bounded and increasing function. Then limx→∞ f(x)
exists. (The whole integral corresponds to the function f here.)
