Indefinite Integrals ilustrated

Consider the definite integrals for the functions 1/x and 1/x^2 on the interval [1, infinity). This applet uses an animated slider to show the "speed" at which 1/x diverges and 1/x^2 converges. The axes have been scaled so that the model contains fairly large numbers. The user should focus the changing yellow numbers, which represent the area under the respective curves, rounded to 15 decimal places.
Can you envision how "long it must take" for the area under the curve 1/x, beginning at left-endpoint a = 1, to reach "infinite area?"