Page 247 - Calculus Workbook For Dummies

231
Chapter 12: Who Needs Freud? Using the Integral to Solve Your Problems

1 /x
x
21. Evaluate lim cscx - logxi. *22. What’s lim 1 + h ? Tip: When plugging
^
_
x " 0 + x " 0
in gives you one of the exponential forms,
Solve It 0 0 ,3 0 , or 1 , set the limit equal to y, take
! 3
the natural log of both sides, use the log
of a power rule, and take it from there.
Solve It

Disciplining Those Improper Integrals

In this section, you bring some discipline to integrals that misbehave by going up,
down, left, or right to infinity. You handle infinity, as usual, with limits. Here’s an inte-
gral that goes up to infinity:

2 4. Integrate.
Q. Evaluate # 1 dx.
x 2 a 2
- 1 = lim - 1 E + lim - 1 E
;
;
A. The area is infinite. a " 0 - x - 1 b " 0 + x b
1. Check whether the function is defined = lim - 1 - - 1 mo + lim - 1 - - 1 mo
e
c
c
e
a
everywhere between and at the limits a " 0 - - 1 b " 0 + 2 b
of integration. = 3 + 3 = 3
You note that when x = 0, the function Therefore, this limit does not exist (DNE).
shoots up to infinity. So you’ve got an
improper integral. In a minute, you’ll see Warning: If you split up an integral in
what happens if you fail to note this. two and one piece equals 3 and the
other equals 3- , you cannot add the two
2. Break the integral in two at the critical to obtain an answer of zero. When this
x-value. happens, the limit DNE.
2 0 2
# 1 dx = # 1 dx + # 1 dx Now, watch what happens if you fail to
x 2 x 2 x 2 notice that this function is undefined at
- 1 - 1 0
x = 0.
3. Replace the critical x-value with con-
2 2
stants and turn each integral into a limit. # 1 dx = - 1 E = - 1 - - 1 m = - 3
c
a 2 - 1 x 2 x - 1 2 - 1 2
= lim # 1 dx + lim # 1 dx
a " 0 - x 2 b " 0 + x 2 Wrong! (And absurd, because the func-
- 1 b
tion is positive everywhere from x = –1
to x = 2.)