Page 289 - Calculus Workbook For Dummies

Chapter 16

Ten Things to Remember

aboutIntegration If You Know

What’sGood for You

In This Chapter
Three approximation rules
The Fundamental Theorem of Calculus
Definite and indefinite integrals and antiderivatives

n this chapter, I give you ten things you should know about integration. If you want to
Ibecome a fully integrated person (as opposed to a derivative one), integrate these inte-
gration rules and make them an integral part of your being.

The Trapezoid Rule

The trapezoid rule will give you a fairly good approximation of the area under a curve in the
event that you’re unable to — or you choose not to — obtain the exact area with integration.
b - a
f x 3 +
T n = 9 f x 0 + 2 _ i 2 _ i 2 _ i ... + 2 _ f x n iC
f x 1 +
f x 2 +
i
f x n 1 + _i
_
2 n -
The Midpoint Rule
An even better area approximation is given by the midpoint rule — it uses rectangles.
b - a x 0 + x 1 x 1 + x 2 x 2 + x 3 x n 1 + x n
M n = n > f d 2 n + f d 2 n + f d 2 n + .......... + d - 2 nH
f

Simpson’s Rule

The best area estimate is given by Simpson’s Rule — it uses trapezoid-like shapes that have
parabolic tops.

b - a
f x 4 +
f x 3 +
f x 1 +
S n = 9 f x 0 + 4 _ i 2 _ i 4 _ i 2 _ i ... + 4 _ f x n iC
f x 2 +
i
_
f x n 1 + _i
3 n -