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Chapter 12: Problem 4
Decide whether the integral is improper. Explain your reasoning. $$ \int_{1}^{\infty} x^{2} d x $$
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Determine whether the improper integral diverges or converges. Evaluate the integral if it converges. $$ \int_{1}^{\infty} \frac{1}{\sqrt[3]{x}} d x $$
Use a spreadsheet to complete the table for the specified values of \(a\) and \(n\) to demonstrate that \(\lim _{x \rightarrow \infty} x^{n} e^{-a x}=0, \quad a>0, n>0\) \begin{tabular}{|l|l|l|l|l|} \hline\(x\) & 1 & 10 & 25 & 50 \\ \hline\(x^{n} e^{-a x}\) & & & & \\ \hline \end{tabular} $$ a=2, n=4 $$
Median Age The table shows the median ages of the U.S. resident population for the years 1997 through \(2005 .\) (Source: U.S. Census Bureau) \begin{tabular}{|l|c|c|c|c|c|} \hline Year & 1997 & 1998 & 1999 & 2000 & 2001 \\ \hline Median age & \(34.7\) & \(34.9\) & \(35.2\) & \(35.3\) & \(35.6\) \\ \hline \end{tabular} \begin{tabular}{|l|c|c|c|c|} \hline Year & 2002 & 2003 & 2004 & 2005 \\ \hline Median age & \(35.7\) & \(35.9\) & \(36.0\) & \(36.2\) \\ \hline \end{tabular} (a) Use Simpson's Rule to estimate the average age over the time period. (b) A model for the data is \(A=31.5+1.21 \sqrt{t}\), \(7 \leq t \leq 15\), where \(A\) is the median age and \(t\) is the year, with \(t=7\) corresponding to 1997 . Use integration to find the average age over the time period. (c) Compare the results of parts (a) and (b).
Use the definite integral below to find the required arc length. If \(f\) has a continuous derivative, then the arc length of \(f\) between the points \((a, f(a))\) and \((b, f(b))\) is \(\int_{b}^{a} \sqrt{1+\left[f^{\prime}(x)\right]^{2}} d x\) Arc Length A fleeing hare leaves its burrow \((0,0)\) and moves due north (up the \(y\) -axis). At the same time, a pursuing lynx leaves from 1 yard east of the burrow \((1,0)\) and always moves toward the fleeing hare (see figure). If the lynx's speed is twice that of the hare's, the equation of the lynx's path is \(y=\frac{1}{3}\left(x^{3 / 2}-3 x^{1 / 2}+2\right)\) Find the distance traveled by the lynx by integrating over the interval \([0,1]\).
Approximate the integral using (a) the Trapezoidal Rule and (b) Simpson's Rule for the indicated value of \(n\). (Round your answers to three significant digits.) $$ \int_{0}^{2} e^{-x^{2}} d x, n=2 $$
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