Question

piecewise function.
integrals.
see below.

Answer 1

the piecewise function f(x) =
\[\frac{ x^2 }{ \left| x \right| }, x \ne 0\]
\[0, x=0\]
find \[\int\limits_{1}^{4} f(x)dx\]
how would u do that?

Answer 2

Riemann integrals are not affected by singular points. So you can do the integral as though the point x=0 doesn't exist.

Answer 3

Plus, the point x=0 doesn't even come up in the integral, since you're taking the integral from 1 to 4.

Answer 4

and so then you know also that x > 0, so on the integral (1,4) f(x) = x^2 / x. (explicitly: you can take away the absolute value sign)

Answer 5

so basically
\[\int\limits_{1}^{4} \frac{ x^2 }{ \left| x \right| } dx\]
?

Answer 6

Yeah that's right.

This is intuitively why singular points don't really matter:
In this picture, there's a singular point where the function is different at just one point. But, you remember when you do the Riemann approximation of the integral (the area under the curve) how you multiply the value of the function by the width? Well Singular points basically have no width... so the area underneath them, is effectively nothing. So you can ignore them.