Finite and Infinite Areas Examples

Given the graph below, does

converge or diverge?

We know

converges, which means the region between the graph of  and the x-axis on [1,∞) has finite area. Call this region B.

Since the region between the graph of f ( x ) and the x-axis on [1,∞) is contained within region B, its area must also be finite. This means

converges.

Given the graph below, does

converge or diverge?

We know

 diverges, which means the region under  on the interval (0,1] has infinite area. Call this region A.

Since the region under f (x) on that interval is even larger than region A, its area must be infinite also, which means

must diverge.

Given the graph below, does

converge or diverge?

We know

 diverges, which means the area under  on the interval (0, 1] is infinite.

Unfortunately, that doesn't tell us anything about whether the area under f (x) on that interval is finite or infinite.

There's a property for definite integrals that says if f (x) < g (x) for all x in [a, b], then

Knowing something is less than infinity doesn't really tell us anything.