Definite Integral Involving Absolute Value

• Oct 14th 2010, 03:54 PM
blackhug
Definite Integral Involving Absolute Value
I am having trouble understanding the following definite integral:
$\displaystyle \int_{-2}^{\frac{-1 + \sqrt{17}}{2}} 5 - x^2 - |x + 1|$

The way I did it is as follows:
$\displaystyle \int_{-2}^{0} 5 - x^2 + x + 1 + \int_{0}^{\frac{-1 + \sqrt{17}}{2}} 5 - x^2 - x - 1$
from which I got a similar answer to the correct one which looked as follows:
$\displaystyle \int_{-2}^{-1} 5 - x^2 + x + 1 + \int_{-1}^{\frac{-1 + \sqrt{17}}{2}} 5 - x^2 - x - 1$

My question is that due to it being an absolute value surely you have to break the integral up into $\displaystyle \int _{a}^{0} + \int _{0}^{b}$?
• Oct 14th 2010, 04:02 PM
HallsofIvy
Why are you breaking the integral at x= 0? At x= 0, |x+ 1|= |1|= 1 and there is no problem. |a| changes "formula" when a= 0 so |x+ 1| changes when x+ 1= 0 or x= -1. Break the integral into $\displaystyle \int_{-1}^{-1}+ \int_{-1}^{\frac{-1+\sqrt{17}}{2}$.
• Oct 14th 2010, 04:20 PM
blackhug
Oh damn, I was treating |x + 1| as |x| (Doh) thanks for the help!