This page includes general and specific information pertaining to the comprehensive assessment for Calculus II (201-NYB-05). The comprehensive assessment will contribute to the attainment of the Science Program competency:
To this end, students will read an exposition of an historical character and write a (thirty minutes or so) test based on the material in said exposition. The exposition concerns an argument that appears in Archimedes' Quadrature of the Parabola.
Classically, the problem of quadrature is to determine a square whose area is equal to that of a given plane figure. (This is essentially what we do when we find that the area of a plane figure is so many square units.) The quadrature of a rectilineal figure (one bounded by straight line segments) presents no serious difficulties; this (among other things) was known to Archimedes' predecessors. Thus, by Archimedes' time, the quadrature of a given plane figure bounded by curves (some of which may be straight line segments) is effected immediately one constructs a rectilineal figure whose area is equal to (or a non-zero rational multiple of) that of the given figure. It is this that Archimedes accomplishes, for a segment of a parabola, in Quadrature of the Parabola. (A segment of a parabola is a region bounded by a parabola and a straight line with which it shares two points.) It represents an extremely important moment in the history of integration because it is the first successful quadrature of a segment of a conic section, and because it presages certain developments involved in the theory of the definite integral.
Note: The remark in Edwards' book, referred to below, to the effect that earlier mathematicians had determined successfully the area of a segment of an ellipse and an hyperbola is incorrect, as Archimedes himself remarks in a letter introducing Quadrature of the Parabola. (C.H. Edwards has confirmed (in private communication) our (W.B.) suspicion that this is a typo in the original exposition, and that the word successfully should have been unsuccessfully.)
In the argument under discussion, Archimedes draws on certain elementary properties of parabolas (some of which he inherits from his predecessors, and some of which he develops himself), as well as Eudoxus' method of exhaustion. (The use of the term ``exhaust'' to describe the method of Eudoxus is something of an anachronism. It appears to have originated in the seventeenth century, with Gregory Saint-Vincent.) In connection with the latter, Archimedes enunciates a principle, which is a forerunner of what is now known as the Archimedean property of the real number system: that the excess by which the greater of (two) unequal areas exceeds the less can, by being added to itself, be made to exceed any given finite area. (The translation used here is given by T.L. Heath, and is included in the link to Heath's translation of Archimedes' Quadrature of the Parabola below.) The Archimedean property of the real number system is encountered later in this course, during the investigation of sequences and series.
Below is a link to a translation into English (as well as a rendering into modern notation) of Archimedes' original treatise for anyone who would like to see (something closer to) Archimedes' actual argument, or Archimedes' other (mechanical) argument. (Archimedes' argument using the method of exhaustion occupies Propositions 18 through 24, which appear at the end of Quadrature of the Parabola.) There is also included a link to a biography of Archimedes, which is kept at one of the better sites on the Internet for the history of mathematics. Neither of these links is part of the compulsory reading for the comprehensive assessment.
There are two sources for the assigned reading.
Students should attempt the exercises at the end of the online copy. Later in the term, each student will write a (thirty to forty-five minute) test based on the exposition (and which may include modifications of some of the exercises at the end of the online copy). The mark on this test will constitute ten percent of the class mark. It will be given during the twelfth or thirteenth week of classes.
Here are some additional resources:
Text based on text by W Boshuck