Page 184 - Calculus Workbook For Dummies

168 Part IV: Integration and Infinite Series

Finding Area with the Trapezoid

Rule and Simpson’s Rule

To close this chapter, I give you two more ways to approximate an area. You can use
these methods when finding the exact area is impossible. (Just take my word for it that
there are functions that can’t be handled with ordinary integration.) With the trape-
zoid rule, you draw trapezoids under the curve instead of rectangles. See Figure 9-1,
which is the same function I used for the first example in this chapter.

y

Figure 9-1:
Ten trape- 2 lnx
zoids (actu- A B
ally, one’s a 1
triangle, but
it works x
exactly like 1 2 3 4 5 6
a trapezoid).

Note: You can’t actually see the trapezoids, because their tops mesh with the curve,
y = ln x. But between each pair of points, such as A and B, there’s a straight trapezoid
top in addition to the curved piece of y = ln x.

The Trapezoid Rule: You can approximate the exact area under a curve between a and
b
f x dx, with a sum of trapezoids given by the following formula. In general, the
b, # ^ h
a
more trapezoids, the better the estimate.
b - a
+
f x 1 +
f x 3 +
f x 2 +
T n = 9 f x 0 + 2 _ i 2 _ i 2 _ i ... 2 _ f x n iC
f x n 1 + _i
_
i
2 n -
where n is the number of trapezoids, x 0 equals a, and x 1 through x n are the equally-
spaced x-coordinates of the right edges of trapezoids 1 through n.
Simpson’s Rule also uses trapezoid-like shapes, except that the top of each
“trapezoid” — instead of being a straight-slanting segment, as “shown” in Figure 9-1 —
is a curve (actually a small piece of a parabola) that very closely hugs the function.
Because these little parabola pieces are so close to the function, Simpson’s rule gives
the best area approximation of any of the methods. If you’re wondering why you
should learn the trapezoid rule when you can just as easily use Simpson’s rule and get
a more accurate estimate, chalk it up to just one more instance of the sadism of calcu-
lus teachers.
Simpson’s Rule: You can approximate the exact area under a curve between a and b,
b
f x dx, with a sum of parabola-topped “trapezoids,” given by the following
# ^ h
a
formula. In general, the more “trapezoids,” the better the estimate.
b - a
f x 3 +
f x 2 +
f x 4 +
+
f x 1 +
S n = 9 f x 0 + 4 _ i 2 _ i 4 _ i 2 _ i ... 4 _ f x n iC
_
f x n 1 + _i
i
3 n -
1
where n is twice the number of “trapezoids” and x 0 through x n are the n + evenly
spaced x-coordinates from a to b.