## Prof. Brian Harbourne's old exam

- [1] Use the substitution u = 4 - 2x to convert the
integral, from x = 0 to x = 1 of 2x(4 - 2x)
^{1/2} with
respect to x, into an integral in terms of
u only. You do not need to evaluate the integral.
- [2] Let R be the region bounded by the curves
y = x
^{1/2}, y = 2 - x, and y = 0.
Sketch the region R, and express its area in terms
of one or more integrals. You do not need to evaluate the integrals.
- [3] Let R be the region bounded by the curves y = 1, x = 1,
and y = 100/x
^{2}. Sketch the region R, and write
down an integral for the volume of the solid formed by
revolving R about the y-axis. You donot need to evaluate the integral.
- [4] A force of 10 pounds stretches a spring 2 inches. Set upan
integral for the work done in stretching the spring 3 inches
beyond its natural length. You do not need to evaluate the
integral, but indicate what units of work the integral will
evaluate to.
- [5] Baby bear's porridge is 150F when Mother bear sets it out
at 9:00 am. The den is a comfy (for bears) 50F. At 9:10 am,
the porridge is 120F, still too hot for Baby bear, who likes
porridge to be 99F. What time will it be when Baby bear's
porridge is just right? (Assume the porridge cools according
to Newton's Law of cooling.)