# Problem 1

Find

# Problem 2

- 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.

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

[0*,*1] about the *x*-axis.

# Problem 3

- 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 *`*.

- 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.)

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