I have a problem that asks for the $\int_3^8 |g(x)| {\rm d}x$. Instead of a value given for $g(x)$ I was given a graph and told to figure it out geometrically.
Does the absolute value sign mean that I first add all the values and then take the absolute value or do I first take the absolute value and then add?
The difference would be when the $g(x)$ goes below the $x$-axis and I have both positive and negative values.
Absolute Value in an Integral
calculusdefinite integralsintegration
Best Answer
In $\int_3^8 |g(x)| {\rm d}x$, the value which you are integrating is $|g(x)|$. As Hendrix says in a comment, this is always non-negative. As such, based on what integration means, you need to always take the absolute values of anything you're using first and then add those over the region of integration, i.e., $3$ to $8$.
Doing it the other way around would give you the wrong answer if you subtract any values (in particular, the result would be too small). Note doing this would be equivalent to solving $|\int_3^8 g(x) {\rm d}x|$ instead.