Calculus: Analytic Geometry and Calculus, with Vectors

278 Integrals

course, we should try to minimize errors by choosing the heights of the

rectangles in such a way that, in each strip, the area of the set which lies

in the region but outside the rectangle is nearly equal to the area of the

set which lies inside the rectangle but outside the region.

This paragraph introduces the trapezoidal formula

fb[o

(4.94) .f (x) dx = e -f - h + Yn-I +

the derivation of which will help us to understand the much better

formula (4.95) which will appear in the next paragraph. We sepa-

rate the interval a < x < b into n equal subintervals of length h, where

h = (b - a)/n, by points xo, x1,. , xsuch that xo = a, xn. = b, and

xk = xk-1 + h for each k = 1, 2, ,n. As in Figure 4.943, where

n = 4, we let Yk = f(xk) for each k. As an approximation to f x'f(x) dxxo

we use L' L(x) dx, where L(x) = Ax + B and the constants are chosen

such that the graph of L(x) = Ax + B is a line passing through the two

points Po(xo,yo) and Pi(xl,yl). The details of the calculation

(4.941) dx = `o+ ' yo (x- xo) dx = h Yo 2 '

fox,

fZIL()

L

are easily supplied; in case yo and yl are positive, the details are super-

fluous because the quantities are equal to the area of a trapezoid and ele-

mentary geometry shows that the formula is correct. Using (4.941) and

analogous formulas, we see that

(4.942) fxkf (x) dx = ek + hYk-12 Yk,

xkl

where the "error term" Ek will be "relatively small" if the graph of f over

the interval xk_1 < x < xk

is "near" the chord joining

P Pk_1 and Pk. Summing

the members of (4.942)

Y4 h 'd if

a=xo x1

Figure 4.943

x2 x3 x4 = b

glues L. e trapezol a or-

mula (4.94).

_ To derive the trape-

x zoidal formula (4.94), we

began by approximating

f(x) over the interval

xo S x 5 x1 by the function L(x) whose graph is a line passing through

the two points Po and P1. To derive the more useful Simpson formula

b

(4.95)= e+[yo+4y1+2y2+4ya+2y4+ff(x)

+ 4yn-1 + yn],