Activity 6.5.2

In this activity we explore the improper integrals $\int_1^{\infty} \frac{1}{x} \, dx$ and $\int_1^{\infty} \frac{1}{x^{3/2}} \, dx$.

1. First we investigate $\int_1^{\infty} \frac{1}{x} \, dx$.

   1. Use the First FTC to determine the exact values of $\int_1^{10} \frac{1}{x} \, dx$, $\int_1^{1000} \frac{1}{x} \, dx$, and $\int_1^{100000} \frac{1}{x} \, dx$. Then, use your computational device to compute a decimal approximation of each result.

   2. Use the First FTC to evaluate the definite integral $\int_1^{b} \frac{1}{x} \, dx$ (which results in an expression that depends on $b$).

   3. Now, use your work from (ii.) to evaluate the limit given by $$\begin{gathered} \lim_{b \to \infty} \int_1^{b} \frac{1}{x} \, dx \end{gathered}$$.

2. Next, we investigate $\int_1^{\infty} \frac{1}{x^{3/2}} \, dx$.

   1. Use the First FTC to determine the exact values of $\int_1^{10} \frac{1}{x^{3/2}} \, dx$, $\int_1^{1000} \frac{1}{x^{3/2}} \, dx$, and $\int_1^{100000} \frac{1}{x^{3/2}} \, dx$. Then, use your calculator to compute a decimal approximation of each result.

   2. 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 $b$).

   3. Now, use your work from (ii.) to evaluate the limit given by $$\begin{gathered} \lim_{b \to \infty} \int_1^{b} \frac{1}{x^{3/2}} \, dx \end{gathered}$$.

3. Plot the functions $y = \frac{1}{x}$ and $y = \frac{1}{x^{3/2}}$ on the same coordinate axes for the values $x = 0 \ldots 10$. How would you compare their behavior as $x$ increases without bound? What is similar? What is different?

4. How would you characterize the value of $\int_1^{\infty} \frac{1}{x} \, dx$? of $\int_1^{\infty} \frac{1}{x^{3/2}} \, dx$? What does this tell us about the respective areas bounded by these two curves for $x \ge 1$?