The function that can satisfied area under the curve and why area is continue to positive infinity ?

Consider a partition of the interval such that



Within each subinterval, select so that the value of . In other words, is specially chosen to guarantee that is the maximum value of within each corresponding subinterval.

So now we have an upper bound for the integral in terms of a Riemann sum,



with . In the limit, these two will be exactly equal:



The main point here is that the right hand side is the sum of the terms of a convergent sequence as .

But diverges if , so the integral must also diverge.

Anyone can me with these answers