Practice Exam: Numerical Integration, Improper Integrals, Applications
Time: 60 minutes

Problem 1 (15 points)   Compute the exact value of $\displaystyle \int_0^\infty x e^{-2x^2}\,dx $

Problem 2 (15 points)   To approximate the integral $\displaystyle \int_a^b f(x)\,dx $, the interval [a,b] is divided into n parts of length $\Delta x$ each. Explain why

\begin{displaymath}\mbox{TRAP}(n)=\mbox{LEFT}(n)+\frac{1}{2}(f(b)-f(a))\cdot\Delta x\end{displaymath}

Problem 3 (20 points)   Do the following integrals converge or diverge? For full credit, you must explain how you arrive at your answer.
$\displaystyle \int_{0}^\infty \frac{x^2-4x+7}{(x^2+6)^2}\,dx $

$\displaystyle \int_3^\infty \frac{(x-2)\ln x}{x^2}\,dx $

Problem 4 (15 points)   For p>0 consider the improper integral

\begin{displaymath}\int_{0}^1 \frac{dx}{x^p}.\end{displaymath}

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.)

Problem 5 (15 points)   Let a,b>0. Find the volume of the ellipsoid you obtain, when you rotate the ellipse $\displaystyle \frac{x^2}{a^2}+\frac{y^2}{b^2}=1 $ about the y-axis.

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 km3).

If you would like to check your answers, click on Answer.

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Helmut Knaust