Activity 6.5.2.
In this activity we explore the improper integrals \(\int_1^{\infty} \frac{1}{x} \, dx\) and \(\int_1^{\infty} \frac{1}{x^{3/2}} \, dx\text{.}\)
(a)
First we investigate \(\int_1^{\infty} \frac{1}{x} \, dx\text{.}\)
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Use the First FTC to determine the exact values of \(\int_1^{10} \frac{1}{x} \, dx\text{,}\) \(\int_1^{1000} \frac{1}{x} \, dx\text{,}\) and \(\int_1^{100000} \frac{1}{x} \, dx\text{.}\) Then, use your computational device to compute a decimal approximation of each result.
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Use the First FTC to evaluate the definite integral \(\int_1^{b} \frac{1}{x} \, dx\) (which results in an expression that depends on \(b\)).
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Now, use your work from (ii.) to evaluate the limit given by\begin{equation*} \lim_{b \to \infty} \int_1^{b} \frac{1}{x} \, dx\text{.} \end{equation*}
(b)
Next, we investigate \(\int_1^{\infty} \frac{1}{x^{3/2}} \, dx\text{.}\)
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Use the First FTC to determine the exact values of \(\int_1^{10} \frac{1}{x^{3/2}} \, dx\text{,}\) \(\int_1^{1000} \frac{1}{x^{3/2}} \, dx\text{,}\) and \(\int_1^{100000} \frac{1}{x^{3/2}} \, dx\text{.}\) Then, use your calculator to compute a decimal approximation of each result.
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Use the First FTC to evaluate the definite integral \(\int_1^{b} \frac{1}{x^{3/2}} \, dx\) (which results in an expression that depends on \(b\)).
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Now, use your work from (ii.) to evaluate the limit given by\begin{equation*} \lim_{b \to \infty} \int_1^{b} \frac{1}{x^{3/2}} \, dx\text{.} \end{equation*}
(c)
Plot the functions \(y = \frac{1}{x}\) and \(y = \frac{1}{x^{3/2}}\) on the same coordinate axes for the values \(x = 0 \ldots 10\text{.}\) How would you compare their behavior as \(x\) increases without bound? What is similar? What is different?
(d)
How would you characterize the value of \(\int_1^{\infty} \frac{1}{x} \, dx\text{?}\) of \(\int_1^{\infty} \frac{1}{x^{3/2}} \, dx\text{?}\) What does this tell us about the respective areas bounded by these two curves for \(x \ge 1\text{?}\)
