Practice Exam: Numerical Integration, Improper Integrals, Applications

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

$\begin{eqnarray*}\int_0^\infty x e^{-2x^2}\,dx&=&\lim_{N\to\infty}\int_0^N x e^{... ...m_{N\to\infty}-\frac{1}{4}\left(e^{-2N^2}-e^0\right)=\frac{1}{4} \end{eqnarray*}$

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

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

$\begin{displaymath}\mbox{LEFT}(n)=f(a) \Delta x+\sum_{k=1}^{n-1} f(x_k) \Delta x.\end{displaymath}$

$\begin{displaymath}\mbox{RIGHT}(n)=\sum_{k=1}^{n-1} f(x_k) \Delta x +f(b) \Delta x.\end{displaymath}$

$\begin{displaymath}\mbox{RIGHT}(n)=\mbox{LEFT}(n)-f(a)\Delta x+f(b) \Delta x.\end{displaymath}$

$\begin{eqnarray*}\mbox{TRAP}(n)&=&\frac{1}{2}\mbox{RIGHT}(n)+\frac{1}{2}\mbox{LE... ...ox{LEFT}(n)\\ &=&\mbox{LEFT}(n)+\frac{1}{2}(f(b)-f(a))\Delta x. \end{eqnarray*}$

Problem 3 (20 points) Do the following integrals converge or diverge? For full credit, you must explain how you arrive at your answer.

1.

$\displaystyle \int_{0}^\infty \frac{x^2-4x+7}{(x^2+6)^2}\,dx$

Solution: The integrand is continuous for all $x\geq 0$ , consequently the only "impropriety" occurs at $\infty$ . For large x-values,

$\begin{displaymath}\frac{x^2-4x+7}{(x^2+6)^2}\approx\frac{1}{x^2}.\end{displaymath}$

(You may use the Limit Comparison Test for a more formal argument.) Since $\displaystyle \int_0^\infty \frac{dx}{x^2}$ converges at infinity by the p-test, so does the integral in question.

2.

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

Solution: The integrand is continuous for all $x\geq 3$ , consequently the only "impropriety" occurs at $\infty$ . For large x-values,

$\begin{displaymath}\frac{(x-2)\ln x}{x^2}\approx\frac{x \ln x}{x^2}=\frac{\ln x}{x}.\end{displaymath}$

(You may use the Limit Comparison Test for a more formal argument.)

For $x\geq 3$ ,

$\begin{displaymath}\frac{\ln x}{x}\geq \frac{1}{x},\end{displaymath}$

and thus by the Comparison Test, the integral in question is divergent, since $\displaystyle \int_3^\infty\frac{dx}{x}$ diverges.

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.)

$\begin{displaymath}\int_{0}^1 \frac{dx}{x^p}=\lim_{t\to 0^+}\int_{t}^1 \frac{dx}... ...left.\left(\frac{1}{1-p}x^{1-p}\right)\right\vert _{x=t}^{x=1}.\end{displaymath}$

$\begin{displaymath}\int_{0}^1 \frac{dx}{x^p}=\lim_{t\to 0^+}\int_{t}^1 \frac{dx}{x}=-\lim_{t\to 0^+}\ln x=\infty,\end{displaymath}$

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.

$\begin{displaymath}\int_{-b}^{b}\pi a^2 \left(1-\frac{y^2}{b^2}\right)\,dy=\frac{4}{3}\pi a^2 b.\end{displaymath}$

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).

$\begin{displaymath}\sum_{h=0}^{h=1.5} (30-20h)(50-20h) \Delta h,\end{displaymath}$

$\begin{displaymath}\int_{0}^{1.5} (30-20h)(50-20h)\,dh=900 \mbox{ km}^3.\end{displaymath}$