Chapter 7.8, Problem 52ES

Suppose that $f$ and g are continuous functions and that

$0\le f(x)\le g(x)$

if $x\ge a.$ Give a reasonable informal argument using areas to explain why the following argument using areas to explain why the following results are true.

(a) If ${\displaystyle {\int }_a^{+\infty }f(x)}dx$ diverges, then ${\displaystyle {\int }_a^{+\infty }g(x)}dx$ diverges.

(b) If ${\displaystyle {\int }_a^{+\infty }g(x)dx}$ converges, then ${\displaystyle {\int }_a^{+\infty }f(x)dx}$ converges and ${\displaystyle {\int }_a^{+\infty }f(x)dx}\le {\displaystyle {\int }_a^{+\infty }g(x)dx.}$

[Note: The results in this exercise are sometimes called comparison tests for improper integrals.]