A. Newton-Cotes Formulæ
a. Graphs
Graphically, a definite integral is the area between the x-axis and the curve f(x). Areas below the axis are negative; areas above the axis are positive.
b. Trapezoids
The area “under” the curve might be approximated most simply by a series of trapezoids and triangles.
Notice that x1 = a and that x8 = b.
c. Interpolating polynomial
In effect, we are replacing the integrand, f(x), by a straight line between each pair of points:
 .
This can be checked by integrating p1(x) analytically.
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