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Chapter 6: Problem 2
Decide whether the integral is improper. Explain your reasoning. $$ \int_{1}^{3} \frac{d x}{x^{2}} $$
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Determine whether the improper integral diverges or converges. Evaluate the integral if it converges. $$ \int_{-\infty}^{\infty} x^{2} e^{-x^{3}} d x $$
Use the Trapezoidal Rule and simpson's Rule to approximate the value of the definite integral for the indicated value of \(n\). Compare these results with the exact value of the definite integral. Round your answers to four decimal places. $$ \int_{4}^{9} \sqrt{x} d x, n=8 $$
Capitalized cost Find the capitalized cost \(C\) of an asset (a) for \(n=5\) years, (b) for \(n=10\) years, and (c) forever. The capitalized cost is given by $$ C=C_{0}+\int_{0}^{n} c(t) e^{-r t} d t $$ where \(C_{0}\) is the original investment, \(t\) is the time in years, \(r\) is the annual interest rate compounded continuously, and \(c(t)\) is the annual cost of maintenance (measured in dollars). [Hint: For part (c), see Exercises \(35-38 .]\) $$ C_{0}=\$ 650,000, c(t)=25,000, r=10 \% $$
Explain why the integral is improper and determine whether it diverges or converges. Evaluate the integral if it converges. $$ \int_{3}^{4} \frac{1}{\sqrt{x-3}} d x $$
Use a spreadsheet to complete the table for the specified values of and to demonstrate that $$ \lim _{x \rightarrow \infty} x^{n} e^{-a x}=0, \quad a>0, n>0 $$ $$ \begin{array}{|c|c|c|c|c|}\hline x & {1} & {10} & {25} & {50} \\ \hline x^{n} e^{-a x} & {} & {} & {} & {} \\ \hline\end{array} $$ $$ a=\frac{1}{2}, n=5 $$
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