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How to handle aboslute values in integrals

  • Thread starter wildman
  • Start date
26
3
1. Homework Statement
My question is how to handle absolute values in integrals. For instance I had this in my homework today:
[tex] \int_{-10}^{10} |t|e^{-2|t|}dt [/tex]


2. Homework Equations



3. The Attempt at a Solution

The answer to the problem without absolute values would be easy given that it is in the integral table... If it were integrated from 0 to 10 it would be easy also. I would just take away the absolute value signs and integrate from -10 to 10. But what do you do when it already is from -10 to 10?
 

Answers and Replies

146
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Remember that a definite integral is a measure of area under a curve.

If you look at [itex]x^3[/itex] from -a to a, the total area would be 0, as due to the symmetry, the positive and negative areas cancel out.

But what if we had [itex]|x|^3[/itex]? Then it'd look the same, except all negative values would be flipped up to the positive. What now? Well, in this example, the area from -a to a is what you want. But due to the symmetry, the area from -a to 0, and from 0 to a, would be identical, so you can just figure out one half of it, then multiply.

Try to see what your graph looks like, and see if this helps you.
 
HallsofIvy
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1. Homework Statement
My question is how to handle absolute values in integrals. For instance I had this in my homework today:
[tex] \int_{-10}^{10} |t|e^{-2|t|}dt [/tex]


2. Homework Equations



3. The Attempt at a Solution

The answer to the problem without absolute values would be easy given that it is in the integral table... If it were integrated from 0 to 10 it would be easy also. I would just take away the absolute value signs and integrate from -10 to 10. But what do you do when it already is from -10 to 10?
I hate to state the obvious: split the problem into integrating from -10 to 0 and from 0 to 10! In the integral from -10 to 0, replace |t| with -t and in the integral from 0 to 10, replace |t| with t.

Of course, as Goldenwind said, you don't really have to do both integrals. Because of the symmetry, the two integrals must be the same. Integrate from 0 to 10, with |t| replaced by t, and double.
 
26
3
Thanks guys!
 

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