I'am reading Tom Apostol's Calculus volume-1 text (page 3 and 4),where he talks about calculating the area under a curve which eventually leads to the concept of the definite integral.In the below figure he chooses an arbitrary point on the base and denote's it's distance from $0$ by $x$.
How can the vertical distance be $x^2$?In particular,if the length of the base itself is $b$,then how come the altitude is $b^2$?
He then winds up by concluding that the area $A = \frac{b^3}{3}$ by considering approximations from above and below.
I have a problem in understanding his derivation due to unfamiliarity with mathematical induction and other rigour.Is there an informal way (i mean less rigour) of computing the area? I'll be very happy if i understand this cause i want to think and solve this problem the archimedes way!!
I'am aware of possible duplicate thread however this is purely based on apostol's text.I even went through this Area under a curve is an integral but couldn't understand it.