converge or diverge? If it converges, what does it converge to?
Look at the sequence of integrals
as b gets larger:
These values are getting closer and closer to 1.
As b approaches ∞,
approaches 1. This means
converges to 1.
Each integral in the above exercise is the area between the graph of and the x-axis on an interval with left endpoint 1.
As b approaches ∞, the area
approaches the total area between the graph of
and the x-axis on the interval [1,∞).
In symbols, the total area between and the x-axis on [1,∞) is
We abbreviate this limit by writing