Proof of Exponential Process Doubling Interval

Task It has been observed that the annual consumption rate of certain nonrenewable resources, e.g. oil, has been growing. Use integral calculus to prove that the amount consumed in any doubling interval exceeds (or is approximately equal to) the entire amount consumed from the beginning of time (t = 0) to the start of the doubling interval.
Moving the sliders of k and r_0 and comparing the values of R_B and R_D, the objective of the task can be achieved.