How do I find the definite integral of an absolute value function?
For instance: $f(x) = |-2x^3 + 24x|$ from $x=1$ to $x=4$
How do I find the definite integral of an absolute value function?
For instance: $f(x) = |-2x^3 + 24x|$ from $x=1$ to $x=4$
Find the roots (thereby splitting the function into intervals on which it doesn't change sign), and in each interval evaluate the relevant function (+f or −f).
In your example, we'll take $f(x) = -2x^3+24x$, so
$f(x) = 2x(-x^2+12) = -2x(x-\sqrt{12})(x+\sqrt{12})$
$\int_1^4 |f| = \int_1^{\sqrt{12}}|f| + \int_{\sqrt{12}}^4 |f| = \int_1^{\sqrt{12}} f + \int_{\sqrt{12}}^4 -f$
I'm sure you can fill in the rest.