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 x–y–z 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. Hook’s 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.)