§6.5 Improper integrals

Introduction

Improper Integrals Involving Unbounded Intervals

Convergence and Divergence

A company with a large customer base has a call center that receives thousands of calls a day. After studying the data that represents how long callers wait for assistance, they find that the function $p(t) = 0.25e^{-0.25t}$ models the time customers wait in the following way: the fraction of customers who wait between $t = a$ and $t = b$ minutes is given by

.

Use this information to answer the following questions.

Determine the fraction of callers who wait between 5 and 10 minutes.

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

• 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. - Use the First FTC to evaluate the definite integral $\int_1^{b} \frac{1}{x} \, dx$ (which results in an expression that depends on $b$). - Now, use your work from (ii.) to evaluate the limit given by

.

Next, we investigate $\int_1^{\infty} \frac{1}{x^{3/2}} \, dx$. - 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. - 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 $b$). - Now, use your work from (ii.) to evaluate the limit given by

.

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?

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 $x \ge 1$?

$\int_1^{\infty} \frac{1}{x^2} \, dx$

$\int_0^{\infty} e^{-x/4} \, dx$

$\int_2^{\infty} \frac{9}{(x+5)^{2/3}} \, dx$

$\int_4^{\infty} \frac{3}{(x+2)^{5/4}} \, dx$

$\int_0^{\infty} x e^{-x/4} \, dx$

$\int_1^{\infty} \frac{1}{x^p} \, dx$, where $p$ is a positive real number