Improper integrals are where the interval is infinite (at either both points in the interval or just one) OR where the function becomes undefined/infinite somewhere in the interval

Types of Improper Integrals

Type 1 - Infinite Interval

• You cannot just plug in infinity, as that is basically asking someone to find the area/volume of something from 1ft to infinite ft, which just is not possible
• Instead of plugging in infinity, we replace it with a limit
• If both sides are infinite we split it into two separate integrals

Type 2 - Discontinuity / vertical asymptote

• Usually when there is a rational function and there is obviously and asymptote where the function blows up in the interval
• Replace the “bad” point or point of issue with a limit

Convergence vs Divergence

Convergence is when the integral approaches a finite number. This is when you find the limit and it exists and equals some finite value.

Divergence is when the integral does NOT approach a finite value, and finding the limit gets you infinity/DNE

• Think the d in diverge = DNE, (the limit DNE)

How To