MTH 4640 – History of Mathematics;Archimedes – Quadrature of a Parabola.
Archimedes used the “method of exhaustion” to find the area of the segment formed by drawing any chord of a parabola. For
details, see pages 197-198 of our text (and the work we did in class). The interesting conclusion of his work is that the area of the
segment is equal to 4/3 the area of the first triangle constructed in his algorithm. That first triangle has the chord as a base, and the
third vertex* is on the point of the parabola where the tangent line to the parabola has slope equal to the slope of the chord.
In part (a) you will confirm, using modern methods (i.e., Calculus), that this result holds for a particular parabola and a particular
chord.
In part (b) you will confirm that this process is well-defined. That is, you will show that with any chord and any parabola, there
will be exactly one x-value that meets the requirements for the third vertex of the triangle (described above and noted with an
asterisk).
In part (c) you will explore the phenomenon that we have observed so far with this procedure. Is it true that the third vertex of the
triangle always has x-coordinate equal to the average of the two endpoints of the chord?
The area result, in symbolic form is: Asegment = 3
4 Atriangle.
(a) Show that the area of the segment formed by the parabola y = x2 and the chord defined by the line y = – ½ x +3 agrees
with the result above. Give exact answers (no rounded decimals) and show all work done by hand. [See the graph given
below.]
(b) Show that with any chord and any parabola, there will be exactly one x-value that meets the requirements for the third
vertex of the triangle.
[Suggestion: Consider a generic quadratic function f (x) = ax2 + bx +c, and a chord between two of its points, say
(p, f (p)) and (q, f (q)), where p is not equal to q. Use calculus and algebra, treating only x as the variable, and see what
happens in using calculus and algebra to find the x-coordinate of the third vertex of the triangle in terms of the constants
a, b, c, p, and q.]
(c) Decide if the x-coordinate of the third vertex is always exactly halfway between the x-coordinates of the endpoints of the
chord. If yes, then prove your result. If no, then give a specific counterexample (by naming a specific parabola and a
specific chord of that parabola).
[Thoughts for part (c):
How does the slope of the chord compare with the slope of the tangent line at x = !!!
! ? It is possible that your work for
(b) will make part (c) trivial.]
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