why is area under 1/x^2 as x approaches infinite finite while area under 1/x is infinite?

Question

They both get closer to but never reaching zero. Is the reason simply that one gets 1/x^2 gets closer to zero faster? So whats the threshold for something to be considered finite or infinite?

In: 59

Answer 1

The answer comes from calculus. We can find a function to represent the area under the curve, called the integral. We can look at the integral and see if it keeps getting bigger or if it eventually stops.

1/x integrates to ln|x|+c. We can see that ln|x| just keeps getting bigger as x gets bigger – there’s no limit where it stops.

1/x^2 integrates to -1/x + c which approaches c as x increases. Measuring the area starting from x=1 to x=inf, we get -1/inf+c – -1/1 – c which simplifies down to 1.