This little exploration, to be done in a small group of students, to lead them into the topic of Improper Integrals. They see if the area under a curve diverges or converges by substituting progressively larger values in the upper bound of the integral. Then, as a last step, they generalize by writing the improper integral as a formal limit as b approaches infinity.
The kids use finite values in a definite integral, observe a pattern in those values, then write their idea formally.
Thanks to master teacher Peter Mili for suggesting I put in the formal last step. Here is the link to the Word document format of the exploration. It’s about 3 pages long. ImproperIntegrals