Integral of an absolute value function

Question

How do I find the definite integral of an absolute value function?

For instance: $f(x) = |-2x^3 + 24x|$ from $x=1$ to $x=4$

Answer 1

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.