Page 182 - Calculus Workbook For Dummies

166 Part IV: Integration and Infinite Series

*11. Use sigma notation to express an 8-right- *12. Using your result from problem 11, write
rectangle approximation of the area a formula for approximating the area
2
under g x = 2 x + 5 from 0 to 4. Then under g from 0 to 5 with n rectangles.
^ h
compute the approximation.
Solve It
Solve It

Close Isn’t Good Enough: The Definite

Integral and Exact Area

Now, finally, the first calculus in this chapter. Why settle for approximate areas when
you can use the definite integral to get exact areas?

The exact area under a curve between a and b is given by the definite integral, which is
defined as follows:

b n
f x dx =
# ^ h lim!= f x i $ c b - a mG
i
_
n
n " 3 i 1
=
a
In plain English, this simply means that you can calculate the exact area under a curve
between two points by using the kind of formula you got in Step 10 of the previous
example and then taking the limit of that formula as n approaches infinity. (Okay, so
maybe that wasn’t plain, but at least it was English.)
The function inside the definite integral is called the integrand.