Practice Exam: Numerical Integration, Improper Integrals, Applications Time: 60 minutes
Problem 1 (15 points)
Compute the exact value of
Solution: The integrand is continuous for all x, consequently the only "impropriety" occurs at .
In the last step we use the fact that
.
Problem 2 (15 points)
To approximate the integral , the interval [a,b] is divided into n parts of length each.
Explain why
Solution: Let
denote the endpoints of the n subintervals.
Then
On the other hand,
Consequently,
It follows that
Problem 3 (20 points)
Do the following integrals converge or diverge? For full credit, you must explain how you arrive at your answer.
1.
Solution: The integrand is continuous for all ,
consequently the only "impropriety" occurs at .
For large x-values,
(You may use the Limit Comparison Test for a more formal argument.) Since
converges at infinity by the p-test, so does the integral in question.
2.
Solution: The integrand is continuous for all ,
consequently the only "impropriety" occurs at .
For large x-values,
(You may use the Limit Comparison Test for a more formal argument.)
For ,
and thus by the Comparison Test, the integral in question is divergent, since
diverges.
Problem 4 (15 points)
For p>0 consider the improper integral
Derive the p-test for these integrals, i.e., find the values of p for which this integral converges, and the values for which it diverges. (To obtain full credit, you must show all steps necessary to obtain your answer.)
Solution: For ,
we have
Since
for p<1, while
for p>1, the integral converges for p<1 and diverges for p>1.
What if p=1? Then
so the integral diverges.
Problem 5 (15 points)
Let a,b>0. Find the volume of the ellipsoid you obtain, when you rotate the ellipse
about the y-axis.
Solution: Rotating the ellipse about the y-axis amounts to rotating the graph
of
about the y-axis.
Each horizontal slice is a disk with area
.
Consequently the volume of the ellipsoid is given by
Problem 6 (20 points)
A rectangular water reservoir is 50 km long and 30 km wide. The depth of the reservoir at each point is one tenth its distance to the nearest shore line. Find the volume of water in the reservoir (in km^{3}).
View from the Top. Darker blue corresponds to deeper water.
Solution: The maximal distance from any point in the reservoir to the nearest shoreline is 15 km; thus the maximal depth of the reservoir is 1.5 km.
The distance to the shoreline is constant on lines parallel to the shorelines.
More precisely, the water is at least h km deep inside a rectangle of width 30-20 h km and length 50-20h km.
This leads to the Riemann sum approximation of the volume as
with corresponding integral
View from the Side (upside down). Darker blue corresponds to deeper water.