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Calculus/Absolute value function

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Expert: - 9/5/2009


Question

Integration |x|/x
I am asked to find the definite integral
of |x|/x  -1<=x<=8

The book answ is 7
I find 9

Not sure how to approach this.

Answer

Integral
Questioner: Mark
Country: United States
Category: Calculus
Private: No
Subject: Integral of |x|
Question:

I am asked to find the definite integral
of |x|/x  -1<=x<=8

The book answ is 7
I find 9

Not sure how to approach this.  
-------------------------
Two ways:

1. Look at the picture.  The green is positive area (8 boxes), and the red is negative. (1 box)  

That nets +7.

2. Use the definition.  (I recommend you NOT show your integration by parts to your teacher -- he will probably make some nasty remarks.)

Def. of  |x|:
     | x, when  x >= 0
|x| = |
     |-x, when  x < 0

Now def. of |x|/x :

       | x/x = 1, when  x >= 0
|x|/x = |
       |-x/x = -1, when  x < 0

Split your integral into two parts.  Actually, it becomes an improper integral, but we can discuss that later.  (Later this year, I mean.)
...............
x > 0 part:

{8
|  1 dx = x, from 0 to 8 = 8 (easy)
}0
..............
x < 0 part:

{0
|   -1 dx = -x, from -1 to 0
}-1

= (-(0)) - (-(-1)) = 0 - (+1) = -1
(look at all those minuses.)
...............
Now combine:

{8    {0     {8
|  =  |   +  |
}-1   }-1    }0

I think you get the idea now.

Paul Klarreich

Expertise

All topics in first-year calculus including infinite series, max-min and related rate problems. Also trigonometry and complex numbers, theory of equations, exponential and logarithmic functions. I can also try (but not guarantee) to answer questions on Analysis -- sequences, limits, continuity.

Experience

I taught all mathematics subjects from elementary algebra to differential equations at a two-year college in New York City for 25 years.

Education/Credentials
(See above.)

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