Many students seem to struggle when they see an integral with modulus as they do not know where to begin with. The first thing they should note is that, we cannot evaluate an integral with modulus directly, that means, we must remove (address) the modulus first.

So let’s see how we should approach such questions, considering \int_{a}^{c} |f(x)| dx, we must first know what range of values of x for which f(x) is negative, in this case, let us assume that f(x) \le 0 for a \le x \le b, where b < c.

Then \int_a^c |f(x)| dx = -\int_a^b f(x) dx + \int_b^c f(x) dx

We break the integral up into two parts, adding a negative sign to the integral part for which f(x) \le 0. Students can relate this to reflecting f(x) about the x-axis to make it a positive area.

Students may want to reference this recent ACJC Prelim 2015 Question to see if they can do it.

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