How would you evaluate definite integrals using absolute value?

Question

anonymous · Answer

Before, I continue, I just want to clarify, you're curious how you might evaluate an integral that has an absolute value in it?

anonymous · Answer

yep!

anonymous · Answer

If you're asked to integrate something that has absolute value then, depending on the limits, you may need to consider multiple integrals. Like for example, say we had
$$\int\limits_{-3}^{1}\left| 3x+2 \right|dx$$
By definition of absolute value

|dw:1416295655942:dw|

So we find the values for x in which 3x + 2 is less than 0 or greater than 0. 
$3x+2 < 0 \implies\ x < -2/3$. Therefore we can say that when x < -2/3, |3x + 2| "behaves" like the function -(3x+2) = -3x-2
Thus for x > -2/3, we can say |3x+2| "behaves" like 3x + 2. This allows us to take two separate integrals now. I want an integral for the values in which the function behaves liek -3x-2 and an integral for the remaining valus in which the function behaves like 3x + 2. Doing that, we have this:
$$\int\limits_{-3}^{-2/3}(-3x -2)dx + \int\limits_{-2/3}^{1}(3x+2)dx$$ So I split the integral from -3 to 1 into 2 pieces based on the information gathered above. Now you would just evaluate the two integrals and sum their answers.