How to handle aboslute values in integrals

[wildman]

Homework Statement

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

Homework Equations

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?

 

[Goldenwind]

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 Statement

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

Homework Equations

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.

 

[wildman]

Thanks guys!