JZfa20 Math 1920 Name: ___________________

Improper Integrals

Show all pertinent work to receive credit. Label and scale the axes of all your graphs.

An improper integral involves the computation of an unbounded area.

– integrals involving a horizontally unbounded region:

A f x dx f x dx a

b a

b

? ? ?

??? ?( ) lim ( )

– integrals involving a vertically unbounded region:

A f x dx f x dx b

b

? ?? ? ? ?

( ) lim ( ) 0

1

0

1

These integrals may converge to a finite value, else they are said to diverge:

EXAMPLE 1. A horizontally unbounded region that diverges:

? ?A x

dx x

dx x x b

b

b x x b

b ? ? ? ? ? ?

?

?? ?? ? ?

?? ? ?

1 1

1 1 1lim lim ln lim ln

whereas

A horizontally unbounded region that converges: