SCIE 001 MATHEMATICS ASSIGNMENT 12 (Due 10:00 am Feb. 14, 2014)
There are two parts to this assignment. The first part is on WeBWorK and is due by 10:00 am on Fri. Feb.
14. The second part consists of the questions on this page. This assignment is due by 10:00 am on Fri. Feb.
14. For these questions, you are expected to provide full solutions with complete arguments and justifications.
You will be graded on the correctness, clarity and elegance of your solutions. Your answers must be typeset
or very neatly written. They must be stapled, with your name and student number at the top of each page.
1. (a) Use a Comparison Test for improper integrals to determine whether
Z ∞
1
dx
x
+
x2
1
is convergent or divergent.
(b) Find aR partial fraction decomposition for 1/(x + x2 ) and use this partial fraction decomposition to
∞
write 1 1/(x + x2 ) dx as a sum of two improper integrals. Determine whether these latter two
improper integrals are convergent or divergent. Does this result contradict the result of part (a)?
Explain.
R∞
(c) Evaluate 1 1/(x + x2 ) dx, if it exists.
2. A bucket that has a mass of 2 kg and a rope of linear mass density 0.8 kg/m are used to draw water from
a well that is 50 m deep. The bucket is initially filled with 20 kg of water and is pulled up at a constant
rate of 1 m/s, but water leaks out of a hole in the bucket at a constant rate of 0.1 kg/s. Find the work
done in lifting the bucket (and rope and remaining water) to the top of the well.