# How to handle aboslute values in integrals

1. Homework Statement
My question is how to handle absolute values in integrals. For instance I had this in my homework today:
$$\int_{-10}^{10} |t|e^{-2|t|}dt$$

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?

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

If you look at $x^3$ 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 $|x|^3$? 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
Homework Helper
1. Homework Statement
My question is how to handle absolute values in integrals. For instance I had this in my homework today:
$$\int_{-10}^{10} |t|e^{-2|t|}dt$$

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.

Thanks guys!