How to Evaluate the Absolute Value Integration?

[roam]

1. Evaluate

$$\int_{-1}^{3} \left|x^2 -4\right| dx$$




3. The Attempt at a Solution

This is the first time I'm trying this type of question & I think I need to use the following theorem for such questions;

f is integrable on a closed interval a to b.

$$\int_{a}^{b}f(x)dx = \int_{a}^{c}f(x)dx + \int_{c}^{b}f(x)dx$$

(c; any point in between, no matter how the points are ordered)

Therefore in my problem $$\int_{-1}^{3} \left|x^2 -4\right| dx$$, I choose c = 2

Now since the integrand is absolute value we have two cases:

1) x²-4
2) 4-x²

I'm not which one of the following is correct;

$$\int_{-1}^{2} (4-x^2)dx + \int_{2}^{3}(x^2 -4)dx$$

OR

$$\int_{-1}^{2} (x^2-4)dx + \int_{2}^{3}(4-x^2)dx$$

Which one is correct and why?

Thanks!

 

[Dick]

|x^2-4| is nonnegative everywhere. So the pair of integrals which have a nonnegative integrand over the region of integration is the correct one. Which is it? The first or the second?

 

[roam]

The first one is non-negative!
I think I understand the idea now, thank you very much!