Problem 1

Find

Problem 2

1. 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 xyz space is given by

Hint: Think about the cross-sectional areas and the perfect symmetry when revolving the function around the x-axis.

1. b) Compute the volume of the solid obtained by revolving the graph of of on

[0,1] about the x-axis.

Problem 3

1. Hooks law states that the force exerted by an ideal spring when extended from its equi-

librium 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 `.

1. Show that

is convergent.

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 lim_x f(x) exists. (The whole integral corresponds to the function f here.)