Consider the parabola y = x2 over the interval [a, b], and let c = (a + b)/2 be the midpoint of [a, b], d be the midpoint of [a, c], and e be the midpoint [c, b]. Let T1 be the triangle with vertices on the parabola at a, c, and b, and let T2 be the union of the two triangles with vertices on the parabola at a, d, c and c, e, b, respectively Figure 14. Continue to build triangles on triangles in this manner, thus obtaining sets T3, T4, . . . . .

a. Show that A(T1) = (b - a)3/8.
b. Show that A(T2) = A(T1)/4.
c. Let S be the parabolic segment cut off by the chord PQ. Show that the area of S satisfies

This is a famous result of Archimedes, which he obtained without coordinates.
d. Use these results to show that the area under y = x2 between a and b is b3/3 - a3/3.

A(S) = A(Ti) + A(T;) + A(T) + = 3.1(%)