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Chapter 12: Problem 24

Determine whether the improper integral diverges or converges. Evaluate the integral if it converges, and check your results with the results obtained by using the integration capabilities of a graphing utility. $$ \int_{0}^{27} \frac{5}{\sqrt[3]{x}} d x $$

Short Answer

Expert verified
The improper integral converges and its value is \(13.5\).

Step by step solution

01

Rewrite the function

First, rewrite the function in a form that allows easier integration. The integrand, \( \frac{5}{\sqrt[3]{x}} \) can be written as \(5x^{-1/3}\).
02

Application of the Power Rule for Integration

Apply the power rule, which states: \[ \int x^{n} dx = \frac{x^{n+1}}{n+1}+C\], where \(n 鈮� -1\). For our function, \[ \int_{0}^{27} x^{-1/3}\ dx = \left[ \frac{3}{2}x^{2/3} \right]_0^{27}\]. We get the expression \( \frac{3}{2} \times (27)^{2/3} - \frac{3}{2} \times (0)^{2/3}\).
03

Evaluation of the Definite Integral

Compute the values to get the definite integral. We find that \((27)^{2/3} = 9\) and \((0)^{2/3} = 0\). So, the integral becomes \( \frac{3}{2} \times 9 - \frac{3}{2} \times 0 = 13.5\). Since the integral is a real number, it converges and its value is 13.5.

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Most popular questions from this chapter

Use a program similar to the Simpson's Rule program on page 906 with \(n=6\) to approximate the indicated normal probability. The standard normal probability density function is \(f(x)=(1 / \sqrt{2 \pi}) e^{-x^{2} / 2}\). If \(x\) is chosen at random from a population with this density, then the probability that \(x\) lies in the interval \([a, b]\) is \(P(a \leq x \leq b)=\int_{a}^{b} f(x) d x\). $$ P(0 \leq x \leq 2) $$

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} $$ a=1, n=1 $$

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. $$ \int_{0}^{2} x^{3} d x, n=8 $$

Determine whether the improper integral diverges or converges. Evaluate the integral if it converges. $$ \int_{1}^{\infty} \frac{1}{\sqrt[3]{x}} d x $$

Lumber Use The table shows the amounts of lumber used for residential upkeep and improvements (in billions of board-feet per year) for the years 1997 through \(2005 .\) (Source: U.S. Forest Service) \begin{tabular}{|l|c|c|c|c|c|} \hline Year & 1997 & 1998 & 1999 & 2000 & 2001 \\ \hline Amount & \(15.1\) & \(14.7\) & \(15.1\) & \(16.4\) & \(17.0\) \\ \hline \end{tabular} \begin{tabular}{|l|c|c|c|c|} \hline Year & 2002 & 2003 & 2004 & 2005 \\ \hline Amount & \(17.8\) & \(18.3\) & \(20.0\) & \(20.6\) \\ \hline \end{tabular} (a) Use Simpson's Rule to estimate the average number of board-feet (in billions) used per year over the time period. (b) A model for the data is $$ L=6.613+0.93 t+2095.7 e^{-t}, \quad 7 \leq t \leq 15 $$ where \(L\) is the amount of lumber used and \(t\) is the year, with \(t=7\) corresponding to 1997 . Use integration to find the average number of board- feet (in billions) used per year over the time period. (c) Compare the results of parts (a) and (b).
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