Classical mechanics, the basis for Newtonian physics, and much of engineering, are founded on and made rigorous by calculus. This is a gateway course to most of the mathematically rigorous intellectual disciplines. At its centre are two perspicuous geometry problems: what straight line segment best approximates a small portion of a given curve and how can one define the area of a region if its boundary is a curve and thus cannot be paved over exactly with rectangular tiles no matter how tiny? Astonishingly enough these problems are not unrelated – roughly speaking each is the 'reverse' of the other, though it takes some time to explain what that means and how it happens.
(Mysterious hint: The word "curve" appears in the statement of each of the problems, but there are two curves under consideration, one for the first problem and a different one for the second.)